That would be true if evidence were a one-to-one relationship but since it's many-to-one, you can't conclusively state !B.

That's the difference between "is evidence of" and "proves". You cannot conclusively state !B, but you can say there is evidence of !B.

Only naively. If there is a clear reason why there is no evidence of absence (e.g. no one has ever looked for evidence), I don't see how this "basic result" is true in any meaningful way.

No, not only naively. Absolutely.

The trick is that the strength of the evidence is not equivalent. It is not correct to say "Evidence of strong evidence is strong evidence of absence". More generally, A is strong evidence of B does not imply that ¬A is strong evidence of ¬B.

Indeed. The alternative would be "we have no evidence that B is happening, therefore we can conclude that it is more likely to be happening than if we had evidence that it was happening."

It is important to consider the chance of us having evidence of B, assuming that B was actually happening.

We have an absence of evidence for æther, but we are pretty certain that if it exists, we would have evidence of it. Our confidence of that is strong enough to state with some confidence that æther does not exist. For practical intents and purposes, we can say that the absence of evidence of æther is evidence that it does not exist.

Not all absences of evidence are made equal, some are more significant than others.

ERM... `a -> b` == `^b -> ^a`. what are you even talking about?

Probability, not logic. See http://lesswrong.com/lw/ih/absence_of_evidence_is_evidence_o...