1. Is there a name for this period?
I came accros this site whilst trying to answer this question for myself. I would agree with other posts that there does not appear to be a formal name. Though the Catholic church appear to name it, "Eastertide" and/or "Easter Season".
2. When He wasn't appearing, where and on what was he spending His time?
Again, this is a question I have tried to research on & off for some time. I would agree with the post above which says words to the effect that, as He had been glorified He was able to move from the heavenlies to earth at will, just as he could move long distances on earth at will, eg from Jerusalem to Tiberias on Galillee.
3. Was His being the same as pre-resurrection, or was He somehow fundamentally different?
Again I must agree with the post above that He was "Flesh & bone" rather than "Flesh & blood". Surely this is a deep mystery. We do know that he was able to eat & drink. He walked about on earth (Road to Emmaus) as humans do which requires a physical frame but His body needed the ability to effectively move through dimensions, i.e. spiritual as well as physical; heaven to earth.
4. Was there a purpose to this "stop-over" back on earth? (I assume to indicate the Resurrection)
The reason I simply added one comment originally is because all of the other questions had been answered very well as had this question. One major point is that Christ gave His disciples the Great Commission during this period (Matt.28: 19,20) which commanded them to take the Gospel to the nations. We read that he also did many miracles duing this period but there is no record of Him preaching the Gospel. One would think this was a magnificent opportunity to show His resurrected body to the world & to work His miracles as He did to the disciples. But it was apparently not in His purpose at this time. Not one conversion is recorded during His Resurrection Ministry.
Of course His resurrection was later preached abroad to all. But He chose to remain hidden to all but His disciples during this time. This is clear from Acts 10: 40,41 & every single reference to His resurrection. In John 7:10 we read of Jesus' last Feast of Tabernacles. In verse 10 we learn that He chose to remain "as it were in secret until the midst of the feast." He stated:
“Yet a little while, and the world seeth me no more; but ye see me: because I live, ye shall live also” (John 14: 19). He continued: “…He that loveth me shall be loved of my Father, and I will love him, and will manifest myself to him“ (verse 21), Again, “Lord, how is it that thou wilt manifest thyself unto us, and not to the world?” (verse 22). So again we note that 3 times in 4 verses we have the clear distinction between the saved (us) and unbelievers (the world).
Isn't this a prophecy of the Lord's Resurrection Ministry? Certainly, He gives at least a partial answer to the purpose of His resurrection ministry; to manifest Himself to believers & not the world. I do hope this an acceptable post