What is the meaning and theme of the 2016 movie "Hail Caesar"?

The Slate.com Review says:

It’s a film about faith, and the pivotal role that it plays in one man’s search for meaning amidst the chaos of existence. It’s a tale of the Christ, told by two Jews who wouldn’t trust Jesus to save a cat out of a tree, let alone someone’s mortal soul.

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When it says that the movie is "told by two Jews", they must mean the Coen brothers.

Hail, Caesar! takes one of the diverse back catalogs in American cinema and forces its various components into a reluctant conversation that changes them all, like strangers who are forced into small talk at a cocktail party only to realize that they have the whole world in common. The Coens tackled the Old Testament with A Serious Man, and with Hail, Caesar! they’ve moved on to the New.

Introduced inside of a dark confession booth, Mannix is starting to doubt the value of his work and the worth of the pictures that he tirelessly endeavors to make possible. Mannix isn’t worried about hits or flops; he’s worried about the cosmic purpose of the film industry, and the existential value of his role in sustaining it. Suffering for the sins of a make-believe world, Mannix is effectively the Jesus Christ of the backlot, and he’s waiting to be crucified at any minute. His faith is being tested. Over the course of 36 hours, each one of which assumes the weight of mortality, Mannix will be forced to find meaning amidst chaos and to believe—truly believe—in something bigger than himself.

The Virgin Mary of this story, Moran is pregnant with the child of a married man, and it’s up to Mannix to defuse the scandal and protect his starlet’s brand of innocence. He knows exactly what to do—he lives for this.
The Coen Brothers’ Hail, Caesar! Is Silly, Epic, and One of Their Very Best

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Is the movie implying that the Virgin Mary secretly had sex out of marriage (raped like in the movie?) and then Jesus was adopted?

The Slate review references the part where Clooney forgets his line as a Roman soldier talking about Christianity "...if only we had faith":

But here’s the thing: Capitol Pictures is a disaster. We only see its films in fragments, but most of those fragments are terrible. Baird Whitlock’s biblical epic looks hammy enough to eat for Christmas dinner

...the value of filmmaking [is argued] to a universe of indelible characters who are struggling to understand it for themselves. It’s a truth they could see if only they had faith. And that, ultimately, is what Hail, Caesar! argues with greater clarity—if not always greater force—than any of the Coens’ previous films. There is no meaning but that which we convince ourselves. It doesn’t matter if you adhere to communism, religion, or movies: The only way you can believe in yourself is if you believe in something bigger.
The Coen Brothers’ Hail, Caesar! Is Silly, Epic, and One of Their Very Best

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The New York Post associates the movie with its debut at Christmas:

Christmas arrives on Friday for fans of classic Hollywood with Joel and Ethan Coen’s “Hail, Caesar!,” which is loaded with (sometimes amusingly scrambled) references to Golden Age actors and their scandals.
One of several films within the film, “Hail, Caesar! A Tale of the Christ” is referred to as a remake. Since the action takes place at an MGM-like studio in the early 1950s, that would make it “Quo Vadis” (1951), to which “Hail, Caesar!” bears plot similarities. But the film’s subtitle belongs to the silent version of “Ben-Hur” (1925), which was remade by MGM in 1959, with yet another version due this summer — and there’s a line of dialogue referring to the films’ famous chariot race.

Baird Whitlock (George Clooney), the fatuous star of “Hail, Caesar!” who’s kidnapped by Communists and converted to their cause, appears to be the Coens’ joke at the expense of the right-leaning stars of both “Quo Vadis” and the 1959 “Ben-Hur.” The former starred Robert Taylor, who testified on Hollywood’s alleged infiltration by Communists before the House Un-American Activities Committee.

And liberal-turned-conservative Charlton Heston won a Best Actor Oscar for “Ben-Hur.’’

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The real-life scandals behind ‘Hail, Caesar!’ | New York Post

Bp. Baron is saying how the movie starts with a crucifix and says Jesus is a theme. He points out that the confession by Mannix is sincere. Ben Hur and The Robe are what Bp. Baron sees as the movies referenced by Hail Caesar.

Bp Baron points out that the end of the movie has another confession scene with Mannix where he deals with tensions. The priest plays the role (allegorically) of the rabbi in A Serious Man, and says You have to do the right thing because God wants you to do the right thing." Yes, he is producing fantasies, but he is doing the right thing or what God wanted, and he gets a burst of confidence and goes back to work. Clooney comes back and says Communist dogma, and slaps him and says Go out there to be a movie star and do the right thing with his life. Then at the end of the movie, the actor delivers a wonderful speech of the purpose of life, and then only says "If only we could have ...." and the answer is faith. Bp. Baron sees this as key, and he needs faith to do that, with God's purpose in your life. Mannix means "man". Mensch in Yiddish refers to the spiritual idea of a good man. Be someone who takes the will and purpose of God seriously. The movie begins and ends with crucifixion of Jesus and with confession, the will and purpose of God.

George Clooney messing up on saying "If only we had faith".

The Atlantic talks about the Rabbi-Priest debate:

The film’s ostensible storyline saw studio executive Eddie Mannix (Josh Brolin) shepherding the production of an epic period drama with the same title, an obvious homage to 1959’s Ben-Hur. In one scene, Mannix gathers four religious leaders—a Protestant minister, a Greek Orthodox patriarch, a Catholic priest, and a Jewish rabbi—to consult them on his film’s portrayal of Christ. “We don’t want to send it to market except in the certainty that it will not offend any reasonable American regardless of faith or creed,” he intones.

The Hail, Caesar! scene quickly ramps up into a theological argument, underlining the absurdity of Mannix’s goal. Immediately, the debate extends beyond the fictional film’s merits and into the nature of Jesus himself. The priest notes that Christ is not God, nor God Christ. “You can say that again! The Nazarene was not God!” the rabbi interjects. “He was not not God,” the patriarch replies. “Part God!” the minister argues. The dialogue bounces around faster and faster as Eddie’s brow furrows: “There is unity in division ... and division in unity,”

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This idea about Jesus being Part God or Christ being not God but the Son of God doesn't sound like either Protestant or Catholic theology, does it? I wonder why they wrote the script that way, then. Some non-mainstream Catholic and Protestant theologians talk that way.

Mannix seems genuinely invested in trying to chart a middle way for his studio’s film, but in the end, he can take satisfaction only in having not obviously offended anyone further. “I haven’t an opinion,” the rabbi ultimately demurs when asked if the film bothered him; after all the angry debate, it’s the perfect laugh-line to end on. The meta-joke, of course, is that much as in Ben-Hur, Hail, Caesar! only depicts Jesus fleetingly, and with no personality outside of a general holy glow.
How 'Hail, Caesar!' Mocked (and Defended) Hollywood in 2016

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Clooney said his model for the movie was Victor Mature, who was in The Robe, rather than Ben Hur or Quo Vadis. I agree that The Robe seems to be the closest film to Hail Caesar.

Neither Ben Hur nor Quo Vadis feature as their protagonist a Roman Soldier who meets Christ at the Crucifixion (Does Ben Hur?).

Christianity Today's article says:

I know there’s no bigger cliché than a Christian critic sitting around identifying “Christ figures” at the movies. But in their latest, Joel and Ethan Coen show their hand so obviously—the subtitle for the Ben Hur-like film-within-a-film, also called Hail, Caesar!, is “A Tale of the Christ”—that I’m either being trolled or baited.

Among many (many, many) things, Hail, Caesar! is a passion play: a canny bit of work on the Coens’ part, given this year’s proliferation of biblical epics both remade and reimagined. In just the next few months, that includes Risen, The Young Messiah, Last Days in the Desert, the Tyler Perry-hosted The Passion Live, and the ABC show Of Kings and Prophets—and, yes, a Ben Hur remake.

Eddie Mannix, the executive in charge of production at Capitol Pictures (that name becomes important later). He goes to confession a lot (“too much,” his priest says wearily) for infractions like smoking a few cigarettes, answers to the never-seen studio head Mr. Schenck (pronounced "harlot"),
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Mannix is watching the dailies (“DIVINE PRESENCE TO BE SHOT,” the subtitles announce at opportune moments—the film is still in production) when he discovers its star has been kidnapped.
'Hail, Caesar!' — A Tale of the Christ?

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Do you think that the last reference "Divine Presence to be shot" has some cryptic meaning, as if it is referring to Clooney as a Christlike figure, since he was kidnapped right after that?

It brings to mind the idea mentioned in the film that Jesus could be God in the sense that the divine presence is in each one of us. I assume that the Coen brothers don't actually take Jesus to be God, and I am skeptical that they would in this film.

Are they implying that just as the Hollywood movies and industry were portraying a false reality and covering up scandals like drunkenness, were the gospels also portraying a fictional, glossed-over narrative, albeit one with morals?

But mostly it’s about the meaning of life by way of religion, with which the Coens have always fiddled, sometimes dancing around the edges and sometimes diving straight into the middle. Hollywood’s Golden Age gives them the perfect excuse for a hysterical scene straight out of a joke: two priests (one Catholic, one Orthodox), a Protestant minister, and a rabbi sit in a boardroom with Mannix, debating whether the depiction of Christ in an upcoming picture “cuts the mustard” or is offensive. As the rabbi points out, for Jews it’s forbidden to portray God, but luckily for them Jesus isn’t part of the godhead. One of the ministers explains that technically Jesus is the Son of God. (The conventional disclaimers at the end of the credits explain that “This motion picture contains no visual depiction of the godhead.”)
... this is a passion play, one with Eddie Mannix at its center, our Man of Sorrows, the savior of the (movie) world. Lest we miss that, the film opens on a long establishing shot of a crucifix before moving to Mannix in the confessional booth, where he’s confessing the most banal of crimes before moving on to his work day.

Unlike every movie executive we’ve ever seen in a film, Mannix is a thoroughly decent guy who speaks nicely to his wife and tries to do his best. But he has reached a crossroads—a point of temptation, if you will. The tempter is a friendly Lockheed Martin executive, who wants him to abandon his true work in the world and come live the easy path.

All day long, Mannix suffers for his stars. He takes their verbal drubbings and deals with their indiscretions and sins and tries to keep them out of trouble, tasked with the thoroughly thankless job of keeping their images squeaky clean.
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Near its end, we catch him in Gethsemane echoes deep in prayer, rosary in hand, as he contemplates what to do—and in a neat trick made possible by the existence of an actual set for a crucifixion scene being shot on the studio lot, he even approaches three crosses on Calvary.
'Hail, Caesar!' — A Tale of the Christ?

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The review concludes that the movie is "a grand mash-up that is then structured like one of the most enduringly popular genres: the biblical epic, the “Greatest Story Ever Told,” the archetypal tale of suffering and redemption."

But the review says not to take this relationship between Mannix and a Christ figure the whole way:

Mannix isn’t the actual man of sorrows; he’s just in the movie business, which is always at its end a bit (or more than a bit) absurd. A speech given by the centurion at the foot of the cross seems like the stand-in for his epiphany—but later he gives a different confession, one that rings more true, about feeling that what he’s doing in the movie business is right and important.

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I am also hesitant to equate him with Jesus because Jesus in the gospels is not portrayed as committing sins, and Mannix's smoking image , mixed with his slapping of Clooney for making criticizims of the film industry and what we know about the negative side of the real life Mannix all goes against the "Christ figure" image.

Another difference is what the film notes about Mannix: "Every day is a fresh set of trials and temptations for the man of sorrows, but he never really faces crucifixion—just another day on set."


Clooney's figure.

To me, it seems that it is Clooney who undergoes the Passion, since he is slapped (by Jolin's character Mannix) for criticizing the film industry's financial decadence (like Jesus slapped by the guard after being arrested for opposing the moneychangers), then drinks a mix that puts him to sleep (like Jesus drinking a mix at the crucifixion after which Jesus died), and then wakes up surrounded by an underground cell of communists (like Jesus showing up surrounded by the underground Christians who were communists).

Is the Coens' movie implying that Jesus was only put asleep by the drink he was given and then woke up later? At one point Joseph Arimathene called Jesus' body a "soma" (sleeping body) when referring to it and talking to Pilate.

The first paranormal or unnatural moment that I noticed was when the Communist study group was relaxing and some of them were putting a puzzle together, and they had put in all the pieces except for the final one, but the final one didn't fit, and they looked at each other quizzically. Now, such a moment is not impossible in nature - the company could have mistakenly sorted the pieces and confused a piece with one in a different set, but that would be extremely rare. It's a very "out of place" moment, and suggests the screenwriter is giving a deeper message.