Yes, but consider what happens when you start conditioning on the statement B="I do not intend to give Omega $5". If Omega is perfect, this is irrelevant; you will hand over the cash.

Right, when Omega is perfect, this isn't really a useful distinction. The correlation between B and A is irrelevant for the odds of p(A|O). It does get more interesting when asking:

p(A|B)
p(~A|B)
p(O|B)

These are still interesting even when Omega is perfect. If, as you suggest, we look at the relationship between A, B, and O when Omega isn't perfect, your questions are dead on in terms of what matters.