Consequences of $SAT \in BQP$
I'm not going to try to answer the first question, as someone like Scott Aaronson, Peter Shor or John Watrous can no doubt give you a far more comprehensive answer on that front.

As regards question 2, it is important to note that quantum computers are in fact more powerful than classical computers in many instances:


There is a rather generic polynomial speed-up gained by quantum computers over classical computers in quite a number of problems. From a complexity point of view, this is perhaps somewhat less interesting than an exponential speed-up, but is something that we can actually prove.
Quantum communication complexity can often vary dramatically from classical communications complexity for the same problem. Again, this is something that can be proven (see for example the Mermin-GHZ game).
Quantum queries to oracles are very often far more powerful than classical queries to the same oracle (see for example the Deutsch-Josza algorithm).


With this is mind, it is already known that quantum computers are fundamentally more powerful than classical computers. I think I would be correct in saying that the majority of physicists who work on such things would already assume that it is not possible to find a classical algorithm to efficiently simulate every quantum system, and so while a result showing that NP was contained in BQP would certainly be surprising, it would not be particularly likely to provide a breakthrough in the understanding of any particular physical phenomenon. Rather it would provide somewhat stronger evidence that quantum physics is hard to simulate.

There is no fundamental physics that is dependent on the computational complexity of simulating it, so finding an efficient quantum algorithm for an NP-complete problem would not have fundamental consequences for the correctness of our current understanding of how the universe functions (though I am inclined to agree with Scott Aaronson's suggestion that it is interesting to see if you could have physical laws emerge from computational assumptions).

It is extremely tempting to say that this would have consequences for adiabatic evolution of quantum systems (and I guess you might get an answer or two suggesting that), etc., but this would be incorrect, as they are governed by a specific physical process, and so showing that it is in principle possible to solve SAT in polynomial time on a quantum computer, wouldn't say anything about their specific evolution.

As regards your last question, we already have examples where problem structure is exploited to yield a polynomial quantum algorithm, but which do not lead to such a classical algorithm (factoring for example). So, as far as our current understanding goes, a problem with a structure exploitable to yield a polynomial time quantum algorithm does not imply that the structure is exploitable to yield a classical polynomial time algorithm.