I think you're oversimplifying the issue.

First, we don't know how long it'll take Harry to revive Hermione. Given the fact that he's made a couple of completely groundbreaking, paradigm-shifting discoveries within a year of being introduced to magic, we have no possible way of predicting how long he'll take to achieve this particular breakthrough. Our own knowledge of magic, of Harry's potential, and of what resources might become available to him in the future, is insufficient to model him in this much depth.

Second, people mature at very different rates depending on their circumstances. Typically, more challenging circumstances make for greater maturity, as long as the person doesn't break down altogether. Being revived after getting murdered and a time-skip has the potential to make Hermione mature a great deal very fast (though she probably won't enjoy it).

Third, Harry's own maturing process so far seems rather non-standard, with rapid growth in some areas and a striking lack of it in others. Again, it is very hard to model what he will become as time goes by, especially given the number of significant character-shaping events that keep getting thrown at him.

Fourth, there is such a thing as friendship across maturity levels. It won't be the same as it was before, but really, we already knew that. Too much was happening to these two from the start for their friendship to maintain any one static form, both in terms of having to respond to external events and in terms of having to learn to deal with each other's somewhat alien ways of seeing the world.

Fifth, what joke is on Harry? He's Harry James Potter-Evans-Verres and his best friend is dead. He's doing it because it's the right thing to do and because she, in particular, matters that much to him, not because he expects things to go back to the way they were.

Or possibly he'll have done enough nasty things "for the greater good" that by the time he gets around to resurrecting her he'll have become someone she can't respect.

Well, Hermione is (well, was) slightly older than Harry, and she seemed to have entered the romantic stage already. A couple years to let Harry catch up might not be such a bad thing.

Hmm, an expensive post. I better make it long then.

First question: I don't think her death is inevitable, I think it has already happened. Author has stated that this story isn't about tricking the readers. So I'm taking her death and her corpses disappearance at face value. Harry has watched her die, watched the time turner limit run out. Her death is as much a part of history as his parent's.

He doesn't even have her brain. His goal isn't her resurrection, if he's being honest, its her re-creation. He has his memories of her. From this, with magic, he intends to once again have a working mind.

Magic can do anything, so this isn't perhaps as preposterous as it seems, but the order of probability of him succeeding at creating a new and identical Hermione seems to be the same whether he does it now or some future wizard gets it done in a century or two as part of a "resurrect everyone who ever died" project.

If he thinks that magic will one day be able to make new people just like those who have died, or time travel beyond the turner's limits, then there's no rush.

Second question: I took it into account, it seemed plausible while he was waiting by her body, still striving to help her. Dumbledore and the other adult's inaction during this time was inexcusable. His frantic efforts to save her were heroic.

But now she's gone. So, what to do?

Well, I figured initially that Harry could dispense with grieving. After all, as his plan is currently, one day he'll make a new Hermione. He presumably won't make her such that she'll miss this period.

But its become clear (to me, that is, this is obviously jmho) that, for Harry, in this instance, declining to grieve has just thrown his emotions out of order. He's...I don't know, flailing seems like the right word. He's doing things that a previous version of him would recognize as anti-rational.

Harry is turning himself into someone Hermione would loathe in order to make a new version of her. He's declaring that he'll vanquish death one moment and then musing on how he'll murder his enemies the next. He's creeping around in an invisibility cloak begging aid from his classmates and then looking down on them for being too dumb to take compensation. (That's unfair for the sake of completing the pattern, but there's actually something way off about someone who doesn't want to be locateable, but still wants the benefits of civilization) He's... I dunno, the word that comes to mind is parody.

Third question: I would agree with you that investigating is much more important than mourning. I disagree that what he's doing is investigating.

His current behavior strikes me as being in line with demanding that his comrades be given all of the magic items, and likely to be similarly effective at preventing any future tragedies his enemy wishes to visit upon him.

I think grieving, if it gave him back his rationality, would be more valuable than what he's currently doing, but less valuable than investigating.

Thanks for reading.