I agree with pretty much everything you've said here, except:

You only cooperate if you expect your opponent to cooperate if he expects you to cooperate ad nauseum.

You don't actually need to continue this chain - if you're playing against any opponent which cooperates iff you cooperate, then you want to cooperate - even if the opponent would also cooperate against someone who cooperated no matter what, so your statement is also true without the "ad nauseum" (provided the opponent would defect if you defected).