Well, there is one important thing that needs to be included first and foremost.
The Ents are not the Children of Ilśvatar. And they are, I believe, even distant from the Dwarves, who are the "Children of Aulė" with the gift of life given to them by Eru. But the Ents do not have the "real life" in this sense, obviously. Therefore, one can start questioning whatever happens to them when they die: I, at least, believe that there is no way they would participate in any "afterlife" in Valinor - and obviously, they do not share the fate of Men, either. I think, if they are close to anything, they are close to animals, or...
...or something else. I believe the key to understanding lies in the Silmarillion, in the chapter "Of Aulė and Yavanna". We hear about the origin of the Eagles and Ents, who are named there together - so in fact, the way I see it, the Eagles are something similar to the Ents. Yavanna speaks about her concern that the trees are defenseless, yet says she did sing about something like that in her part during the Music, and Eru then replies to Manwė that he of course is aware of that, and that
Quote:
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"(...) When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared. For a time: while the Firstborn are in their power, and while the Secondborn are young."
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So here we have it. The way I see it, there are some "spirits from afar" who "incarnate" ("in-wood-ate") into the Ents (and also the Eagles, as the text mentions right after this part) and then dwell in there. Who knows what it is. Some sort of "lesser Ainur"? Some "subthoughts of Eru" created for this particular purpose, who are only later sent to Arda? Who knows.
But in any case, it seems that those spirits are past the simple "mortal dialectic" and whatever happens to them after death is likely leaving their body (tree...) and maybe returning back to Eru or wherever it was they came from.
Personally, I am also wondering whether the Ents fit into a similar cathegory as the Dragons (creations of Morgoth in a similar manner) or the Werewolves, of whom we hear a similar story: "evil spirits" (in this case) placed into a beast's form.
But yes, as for the first question: I think the Ents do "work" that way that they don't age much (remember Treebeard talking about his walks during the First Age), at most they can become "treeish" (maybe the spirit gets "tired" and goes "asleep"?). But of course, they can be killed (by destroying the "corpse", however, the spirit probably simply goes back to its maker again?).