I wasn't wanting to get too far, too quick in sharing with you while you're in the season that you mentioned and perhaps have additional questions once you've seen some answers to your first questions, either from what's already been said or just additional questions you might care to bring up. But this may help as well, and I neglected to post it with my first post.
When you examine the context of each of the times that "Gehenna" comes up in the Scriptures, Jesus is chasing someone over a wrong paradigm, wrong teachings, wrong thinking, wrong beliefs about Him, etc. So, while Matthew 25, 1Corinthians 5, Tartarus in Peter's epistles, and the Lake of Fire seem to be about works, Gehenna is about the usage of your words to others, that James 3 says set on fire the course of nature with the fires of Gehenna.
It's sufficient to prove: the limited duration of each, that the abolishment of death in 1Corinthians 15 would include "the Second Death," and that all of the promises of both destruction AND restoration must be fulfilled so that God will ultimately be all in all, according to 1Corinthians 15:28: not all in a few, or all in an elect, or all in most, but all in all.
But I personally am having growing reservations as to whether there's any afterlife punishments at all, because "age" just means "age" and can be as short as 3 days as in the "forever" that Jonah was in the belly of the fish. "Ages of ages" isn't necessarily much longer than that in it's usage in the Septuignt because Isaiah 34 has two different "forever and ever" judgments that don't overlap, and that happened rather quickly, and that obviously ended ages ago. First the land is given over to fire "forever and ever" and then it's given over to all kinds of weird animals that I guess at the time would have made the land less desireable or perhaps in other ways severely problematic "forever and ever." Yet, it's very clear that both of those judgments ended ages ago.
But the judgments break down into:
Sheol/Hades which are simply the grave;
Gehenna is about false beliefs, false doctrines, wrong paradigms, etc., etc.
Tartarus, Matthew 25, 1Corinthians 5, and the Lake of Fire all seem like judgments related to our works.
And in the Old Testament the way that "afterlife punishments" would often seem to work themselves out would be in people's descendants and what would befall communities after people were gone. The principle of the sins being visited to the third and fourth generation seems to me to be the equivalent of "not forgiven in this age or in the age to come."
That's just where I'm at, though many still believe in afterlife punishments culminating in all being restored to God through Jesus Christ.
Another passage that might occur to you is about not fearing those that can destroy the body only, but to fear him that can destroy both body and soul in Gehenna. The KJV and other translations that by their own admission follow in it's traditions will capitalize the "Him" to fear that can destroy both body and soul in Gehenna, but I believe that the "Him" should not be capitalized because again it's "tradition" on the passage. And Jesus, anticipating the misunderstanding about His Words in future generations, immediately contrasted what He was saying to the crowds about the Pharisees and their doctrines with the love and tenderness of the Father. He immediately said something different about our Father. So although our heavenly Father from the perspective of power, rights, and whatever else "could" destroy both body and soul in Gehenna, there's not much in the New Testament to say that He deliberately would even "temporarily." They were all warnings about what would be the natural outcome of a continued life of disobedience and unbelief, that they "would not enter into His rest" where this life would be concerned. And while they were in this life persisting in unbelief and disobedience, things were just going to go downhill fast!
Inheriting God's kingdom is about inheriting God's Way of dominion, and not an eschatological (i.e. "end-times") event. Romans 5 says that those that have received the abundance of grace and of the gift of righteousness are able to reign in this life through Jesus Christ. Some are inheriting the kingdom, through walking the spiritual way of Jesus Christ rather than through the flesh and blood distinctives of who they were born to or whatever, and some are in danger of not inheriting the Kingdom. If you don't inherit the Kingdom, then you're simply inherited as a part of the Kingdom. And if there are future ages to set additional layers of Creation free, from Hell or from other areas of darkness or areas that they're burning in anxiety, cares, etc., etc., then if you've not inherited then you either might not have much to do during that process or will simply find it much more difficult since this age prepares the way for the next one.