the father, son and holy spirit are three distinct personalities, as distinct as any three people are from each other.
That's right: they exist as three separate Persons at the same time: "Jesus also being baptized, and praying, the heaven was opened, And the Holy Ghost descended in a bodily shape like a dove upon him, and a voice came from heaven, which said, Thou art my beloved Son; in thee I am well pleased" (Luke 3:21-22).
So they are not simply different "modes" that God takes on only one at a time.
They have always existed and have always been distinct.
That's right, except that the Son pre-existed as the Word: "In the beginning was the Word, and the Word was with God, and the Word was God ... And the Word was made flesh, and dwelt among us" (John 1:1,14).
They have all have the same essence. What's the nature of that essence?
The essence of God is Spirit: "God is a Spirit" (John 4:24).
Despite being three distinct personalities, they are one, because of the essence they share.
That's right; they are one in Spirit: "But ye [Christians] are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his" (Romans 8:9).
The Holy Spirit is the Spirit of the Father. Compare the following two passages: "When they shall lead you, and deliver you up, take no thought beforehand what ye shall speak, neither do ye premeditate: but whatsoever shall be given you in that hour, that speak ye: for it is not ye that speak, but the Holy Ghost" (Mark 13:11). "When they deliver you up, take no thought how or what ye shall speak: for it shall be given you in that same hour what ye shall speak. For it is not ye that speak, but the Spirit of your Father which speaketh in you" (Matthew 10:19-20).
At the same time, the Holy Spirit is the Spirit of Jesus Christ: "And I [Jesus] will pray the Father, and he shall give you another Comforter, that he may abide with you for ever; Even the Spirit of truth; whom the world cannot receive, because it seeth him not, neither knoweth him: but ye know him; for he dwelleth with you, and shall be in you. I will not leave you comfortless: I will come to you" (John 14:16-18).
If jesus is the only begotten son of the father, does this mean that the father begat him at some point?
Yes, Jesus is the only-begotten (only-born) Son of God (John 3:16) because he is the only human born without any human father: "Then said Mary unto the angel, How shall this be, seeing I know not a man? And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God" (Luke 1:34-35).
did he need someone else to beget him, like the holy spirit?
As shown in the above passage, Jesus was conceived in the virgin Mary through the operation of the Holy Spirit. This is also referred to elsewhere. For example: "Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost" (Matthew 1:18).
In the bible it says that the son is seated at the right hand of the father, does this mean these beings are subject to space, direction and finity?
No, God's existence is in no way limited: he is omnipresent by his Spirit: "Whither shall I go from thy spirit? or whither shall I flee from thy presence? If I ascend up into heaven, thou art there: if I make my bed in hell, behold, thou art there. If I take the wings of the morning, and dwell in the uttermost parts of the sea; Even there shall thy hand lead me, and thy right hand shall hold me" (Psalms 139:7-10).
But Jesus (the Word, the Son) is also now at the same time in a physical fully-human body: "Behold my hands and my feet, that it is I myself: handle me, and see; for a spirit hath not flesh and bones, as ye see me have" (Luke 24:39).
It is in this fully-human physical body that Jesus ascended into heaven and now sits at the right hand of God the Father as our high priest: "We have such an high priest, who is set on the right hand of the throne of the Majesty in the heavens" (Hebrews 8:1). "This man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God" (Hebrews 10:12).
Jesus will remain in a fully-human body forever so he can serve as the high priest of saved humans forever: "this man, because he continueth ever, hath an unchangeable priesthood. Wherefore he is able also to save them to the uttermost that come unto God by him, seeing he ever liveth to make intercession for them" (Hebrews 7:24-25).
"For verily he took not on him the nature of angels; but he took on him the seed of Abraham. Wherefore in all things it behoved him to be made like unto his brethren, that he might be a merciful and faithful high priest in things pertaining to God, to make reconciliation for the sins of the people. For in that he himself hath suffered being tempted, he is able to succour them that are tempted" (Hebrews 2:16-18).
There is a false doctrine called "antichrist" doctrine, one of the central teachings of which is the ancient Gnostic lie that Christ is not in a fully-human body, not in the flesh: "And every spirit that confesseth not that Jesus Christ is come in the flesh is not of God: and this is that spirit of antichrist, whereof ye have heard that it should come; and even now already is it in the world" (1 John 4:3).
"For many deceivers are entered into the world, who confess not that Jesus Christ is come in the flesh. This is a deceiver and an antichrist. Look to yourselves, that we lose not those things which we have wrought, but that we receive a full reward. Whosoever transgresseth, and abideth not in the doctrine of Christ, hath not God. He that abideth in the doctrine of Christ, he hath both the Father and the Son. If there come any unto you, and bring not this doctrine, receive him not into your house, neither bid him God speed: For he that biddeth him God speed is partaker of his evil deeds" (2 John 1:7-10).
So Christians cannot even welcome into their homes anyone claiming to be a Christian who teaches the lie that Jesus Christ is not in the flesh, not in a fully-human body. This shows how crucial the doctrine of Christ-in-the-flesh is to the true Christian faith.
Because it is so crucial, Satan will no doubt try to completely obliterate the true Christ-in-the-flesh doctrine during the coming world-reign (Revelation 13:7) of the man commonly called the "Antichrist", who "shall come" (1 John 2:18).