1) Yes, the prior is the weighted average of posteriors. This is just the decomposition of P(A) into the sum over b of P(A|b)P(b). The rules applied to do this are the product rule and the mutual exclusivity and exhaustiveness of the different b.

Eliezer has a post on this called "conservation of expected evidence."

2) True, though in anthropic problems this requires more than usual caution, because of the commonness of non-barking dogs (that is, places where you gain information even though no flashing signs pop up to make sure everyone knows you gained information).

In fact, I wrote the above sentence before looking at the blog post. And lo and behold, it's relevant! Allen Downey says:

Whenever SB awakens, she has learned absolutely nothing she did not know Sunday night.

This is not so! But the information gained is what we sometimes call 'indexical' information - information about where, when, or who you are. When you wake up, the thing you learn is that you are now inside the experiment. That seems like a pretty important new thing to know.

I really like Downey's train analogy. The trick, and the way to get ordinary Bayesian reasoning to work here, is to make sure to give different events their own probability - only when you treat the two local trains as two separate events (one way to do this is by setting aside two different labels for them), do you get the right answer. If you just say that P(express train)=1 and P(local train)=1 and stop there, you fail to capture some of your knowledge about the world. You have to say something like P(EXPR)=1, P(LOC1)=1, P(LOC2)=1, P(local|LOC1)=1, P(local|LOC2)=1 - you have to tell the math that being a local train is a property held by two different actual trains.

As for the claim about betting, let alone calling it a Fundamental Theorem, the entire point of the Sleeping Beauty problem is that the bet pays out to a different number of people than actually made the bet before the experiment. Depending on how this is expected to play out, different betting strategies can be right. If all actual transactions only occur at payoff time, though, it seems correct to only consider the situation then.