I don't know much about these issues.
Me neither...
This site lists numerous Eastern rites within the Catholic Church:
https://www.ewtn.com/expert/answers/catholic_rites_and_churches.htm
https://www.ewtn.com/expert/answers/catholic_rites_and_churches.htm
Yes...
What do you mean when you say that the Eastern rite churches were established by the Catholic Church?
They were established in Orthodox countries by the Latin Church through political power as a consequence of invasion and political takeover... The Latin Rite was not recognizable to these countries peoples who had only known the Orthodox Church back to their beginnings, so that the Latins established churches in Communion with the Pope in Orthodox countries with "Eastern Rite" Services, married priests, etc etc. "Real" Latin Priests, celibate and monastic, scorned these 'defiled' priests as a step down, and they were NEVER elevated to positions of power in the Papal hierarchical structure, and none of them ever became a Pope. They were inserted into Orthodox countries in violation of the Canons of the Councils in order to attack the Body of Christ and bring it under subjection to Papal rule... It was a concealing of the Latin Rite in the Eastern "Rite", because when the military withdrew, and the lands returned to their Apostolic Orthodoxy, with that withdrawal departed also the Latin Rite Churches, and what remained was the Latin Communion hiding under the concealment of the appearance of Orthodoxy, but NOT in Communion with the Source of the Rite they were imitating...
The effort failed...
But it created a lot of troubles, and is an enduring source of conflict between the Churches, where the Latins feel justified in punishing the Greek Rebels who have forsaken their obedience to the Pope and his ruling Authority... And the Orthodox see the Latin incursion as a gross violation of the Body of Christ in Orthodox countries...
I don't know much about this at all.
Either do I...
For example, how exactly were the Syro-Malabarese Catholics established?
I am not familiar with these - Perhaps Podromos can fill us in...
As I see it, all the churches were established by Christ through the Catholic Church. All churches within the Catholic Church are Apostolic. Do you mean established directly by the Apostles? Have the churches established directly by the Apostles always been faithful, and never have fallen into heresies like Arianism or Semi-Arianism or Monophysitism or Nestorianism, or other heresies? I don't know which churches were established directly by the Apostles or why that would be the definitive issue.
The Orthodox Catholic Church in Antioch did not establish the Greek Church in Thessalonica, but the Church in Constantinople did establish the Russian Church, and the Serbian Church... They won their autocephaly as they matured from their mother Churches and began to run their own affairs. This is how Churches are spread. The Latin Church did NOT establish the Church in Russia - That was done through Kyrill and Methodios, resulting in the Kyrillic alphabet... The "Eastern Rite Catholic Churches in Communion with (Latin) Rome" are all post-schism Churches established by Latin Rome in countries outside the territories of the Latin Church of the West, in territories legitimately under Orthodox Ecclesiastical responsibility, and are a reflection of the scorn of the Latins for the Orthodox Christian Faith... An extension of the 4th Crusade, because the Eastern Church will NEVER come UNDER Papal Rule... And therefore subject to ANY predations to which the Latin Church subject them...
The reason the issue has relevance is because there developed historically 5 major Apostolic Sees in the Mediterranean area, Alexandria, Jerusalem, Antioch, Greece, and Rome... The Church is hisorically understood as the Communion of these 5... With Constantinople functioning as the capitol of the Roman Empire...
Now Rome went off the rails in asserting Her AUTHORITY OVER the rest of the Apostolic Sees, and NOT ONE of these Sees ever agreed with Her, nor has EVER been under Her authority... And yet She parades Herself on discussion boards as the DEFINING COMMUNION of the Christian Church on earth, when the truth is, by doing so, She has REMOVED Herself from this Communion as Christ established His Body on earth...
My understanding is that the Orthodox believe that Bishops are fallible.
Of COURSE they are fallible - They confess themselves to be the worst of sinners at every Divine Liturgy [joining with Paul, btw], and beg forgiveness from their flock and their concelebrants... AS DO WE ALL... Our Bishops do not strut back and forth parading their false infallibility, but instead confess their sinfulness in humility and lowliness of heart...
Not at all...
I think the definitive issue, ecclesiastically, is whether one is in communion with the Successors of Rock--on whom Christ built the Church and to whom he gave the keys of the kingdom and whose faith he specifically prayed that it would not fail-- the Vicars of Christ.
Christ is the Rock, and He NAMED Peter Petros, Rock, because He confessed Christ as Lord because God the Father had REVEALED to him the identity of Christ. This confession by revelation from the Father is the Rock, not merely the Apostle Peter... The text in Greek shows what the Aramaic would not, that the two rocks are not the same rock, because "this rock" (eg petra) upon which the Church is to be built [the Revelation of the Father] is a feminine noun, and Peter the Rock is "Petros", a very masculine Rock. I should think that your Latin translation of the Greek would reflect the same... It is lost again in English...
Doctrinally, they teach--on behalf of Christ and the Catholic Church--that Mary is immaculately conceived.
ONLY the Latin Communion teaches this, and NONE of the other Churches has EVER taught it, nor will they...
I don't know how this could not be the case, since she is the New Eve.
Even the first Eve was created without the deprivation of holiness.
Certainly the New Eve would be, like the New Adam, free of all sin.
You may rest assured that the Body of Christ knows how...
Arsenios