Logically, "self-defeating" is not equal to "not self-defeating". If the skeptic rejects logic, then he should accept that "self-defeating" is equal to "not self-defeating". Therefore, if logic is self-defeating, then logic is also not self-defeating.
As for the second point - the epistemic perspective is more important than the ontological one. Seriously, read the conclusion of the "Simple truth".
This debate is getting silly, I'm out of here.
On the first point, if you get to a conclusion within logic which marks it as "self-defeating", then from a logical perspective logic doesn't work. Non-logic doesn't matter for those who aren't logical, but for a logical person logic matters.
On the second point, once you start postulating actually viable alternatives to the world not existing, and considering the Evil Demon Argument, there is nothing in there which is actually dealt with.
I've raised arguments for philosophical scepticism before, which have mostly been argued against in a Popper-esque manner of arguing that even if we don't know anything with certainty, we can have legitimate knowledge on probabilities.
The problem with this, however, is how you answer a sceptic about the notion of probability having a correlation with reality. Probability depends upon axioms of probability- how are said axioms to be justified? It can't be by definition, or it has no correlation to reality.