1. Is there a name for this period in the various traditions?
I am not aware of any for protestant Christianity, but Catholicism refers to this period as Eastertide or the Season of Easter.
2. When he wasn't appearing to people, where and on what was Jesus spending his time?
Jesus told the thief on the cross that "today, you will be with me in paradise", so it's reasonable to assume that he was moving between this creation and the next eternal creation being prepared for the redeemed of the Lamb (that place commonly known as "heaven").
That said, neither scripture nor tradition has anything definitive to say about where he was when he was not revealing himself here on earth. Do note, though, that the concept of time is very different outside of this creation (if it has any meaning at all).
3. Was his being the same as pre-resurrection, or was he somehow fundamentally different?
Jesus' body, post-resurrection, was indeed fundamentally different - it was his glorified eternal body, not limited by created time and space. This body is the incorruptible, immortal and eternal state that is the future for all those who are saved, as described in 1 Cor 15:
1 Cor 15:35-49
35 But someone will ask, “How are the dead raised? With what kind of body do they come?” 36 You foolish person! What you sow does not come to life unless it dies. 37 And what you sow is not the body that is to be, but a bare kernel, perhaps of wheat or of some other grain. 38 But God gives it a body as he has chosen, and to each kind of seed its own body. 39 For not all flesh is the same, but there is one kind for humans, another for animals, another for birds, and another for fish. 40 There are heavenly bodies and earthly bodies, but the glory of the heavenly is of one kind, and the glory of the earthly is of another. 41 There is one glory of the sun, and another glory of the moon, and another glory of the stars; for star differs from star in glory.
42 So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. 43 It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power. 44 It is sown a natural body; it is raised a spiritual body. If there is a natural body, there is also a spiritual body. 45 Thus it is written, “The first man Adam became a living being”; the last Adam became a life-giving spirit. 46 But it is not the spiritual that is first but the natural, and then the spiritual. 47 The first man was from the earth, a man of dust; the second man is from heaven. 48 As was the man of dust, so also are those who are of the dust, and as is the man of heaven, so also are those who are of heaven. 49 Just as we have borne the image of the man of dust, we shall also bear the image of the man of heaven.
And this is part of the stunning wonder of the incarnation; that eternal God could have put off his glory to become, for a time, limited, finite man constrained by our dimensions of time and space, to effect the salvation of those who "confess with [their] mouth that Jesus is Lord and believe in [their] heart that God raised him from the dead." (Rom 10:9).
4. Was there a purpose to this "stopover" back on Earth? (I assume to indicate the resurrection).
It was, in part, to bear witness to his bodily resurrection. For as Paul says:
1 Cor 15:12-14
12 Now if Christ is proclaimed as raised from the dead, how can some of you say that there is no resurrection of the dead? 13 But if there is no resurrection of the dead, then not even Christ has been raised. 14 And if Christ has not been raised, then our preaching is in vain and your faith is in vain.
And also that he could give final teaching and instruction to his disciples now that his purpose and mission had been fully revealed to them.