Yes, it was fitting that Christ was baptized (which, of course, was not a Christian baptism), and this explains the why of it well:
“In presenting Himself to receive the baptism of John, Jesus submits to the will of His Father (Mt. 3:14f) and humbly placed Himself among sinners. He is the Lamb of God who thus takes upon Himself the sins of the world (Jn. 1:29, 36). The baptism of Jesus in the Jordan announces and prepares for His baptism ‘in death’ (Ll/ 12:50; Mk. 10:38), which indeed frames His public life between two baptisms. This is what John the Evangelist means when he reports that water and blood flowed from the pierced side of Jesus (Jn. 19:34f) and when he says that the Sprit, the water and the blood are intimately connected (1 Jn. 5:6-8),” Xavier Leon-Dufour, Dictionary of Biblical Theology.
What strikes me in the above is the imagery of Our Lord, the innocent Lamb of God, entering the water in which sinners have entered as a sign of repentance and comes out of the water bearing the weight of those sins - and also cleansing the water, the material that will be used to bring about sinners’ rebirth in Christian baptism (the Church Fathers speak of this).
All this, of course, is not something that the Apostles could accomplish through their own baptisms (for those who were baptized by John, which, again, was not a Christian baptism).
Yes, He did, because they were commissioned to bring others to the rebirth into His Kingdom through the baptism that is the Christian’s immersion in Christ’s death, burial and resurrection, as St. Paul makes clear. Since Pentecost, of course, everybody is normally to be baptized by the sacrament, even St. Paul.
Yes, he was. He required baptism because - unlike the Apostles and the close disciples - he wasn’t present as were the Apostles to all that Baptism signifies. The Twelve were baptized in the experience of the death and resurrection of Our Lord. Thomas was specifically reconciled to Christ the week after by faith in the experience of meeting the Risen Lord. At Pentecost the completion of this occurred for the Twelve gathered in prayer. Garrigou-Lagrange’s classic book, The Three Ways of the Spiritual Life, summarizes a whole tradition of understanding the development of the Twelve’s spirituality from beginners (during the earthly life), the purification of the senses (in the event of the Passion and Death), the infusion of contemplative prayer (in the 40 days of the Resurrection), the purification of the spirit (in the awaiting of the Counselor in the presence of the Blessed Mother drawing down the Holy Spirit), and Pentecost Day (the grace of union or the mystical life) that propels them to preach and incorporate new members from among the Jews through baptism and confirmation.
Yes, it is.