The answer to your question depends upon what you mean by "the asteroids" and "really." The vast majority of the material in the asteroid belt today was probably originally there by the end of solar system formation. Objects in their current size and orbits, not so much.
Size-wise, the asteroid belt has gone through something like 4.5 billion years of collisional evolution, grinding down the larger bodies into the population we see today. It is almost a certainty that the asteroid size-frequency distribution that we observe today is not what it was just after they formed. And, we know this is at least true of asteroid families for which it is possible to compute a model age of when the families formed (the larger object broke up).
Rather than refer you to a drier source for that point, I'll link you to a blog post I wrote on the Baptistina family and its breakup time. The post is actually written mostly by the first-author of a research paper that discusses his original work in light of new work, so it gives you a taste of how this stuff is done more than a wiki article could.
The fate of the asteroid belt is - barring something unexpected - the same as it's been for the last four or so billion years. It should remain pretty much the same as it is now, though slowly grinding down and very slowly losing mass. I don't know what the dynamical timescale of this mass leakage is, but I would suspect it's much longer than the expected lifetime of the sun.
As for your "update" question, it's not really phrased in a way that makes sense: "Are all the objects really still there from the LHB?"
The LHB, by definition, is a bombardment of objects from the asteroid belt in the inner solar system, so those objects would no longer exist. If you mean, did the asteroid belt look different prior to the LHB - if there was an LHB - then the answer is "probably" but there's no good way to tell how. I work with a guy who thinks there was an "E Belt" ("extra") interior to the current Mars-side edge that may have existed before the LHB but been completely depleted by it. The only real argument he can make for it, though, at least at the moment, is that it works dynamically. Finding any sort of unique evidence for it would be pretty difficult.
If that's not what you meant by your question, clarify and I'll try to update my answer.