A premise isn't self-evident because anybody whatsoever would accept it, but because it must be true in any possible universe.
Deductive arguments aren't self-evident, but for a different reason than you think- the Evil Demon Argument, which shows that even if it looks completely solid it could easily be mistaken. There may be some way to deal with it, but I can't think of any. That's why I came here for ideas.
You claim my standards of justification are too high because you want to rule skepticism out- you are implicitly appealing to the fact skepticism results as a reason for me to lower my standards. Isn't that bias against skepticism, lowering standards specifically so it does not result?
There are all kinds of things that are true in every possible universe that aren't self-evident. Look up "necessary a posteriori" for examples. So no, self-evident is not the same as necessary, at least not according to a very popular philosophical approach to possible worlds (Kripke's). More generally, "necessity" is a metaphysical property, and "self-evidence" is an epistemic property. Just because a proposition has to be true does not mean it is going to be obvious to me that it has to be true. Even Descartes makes this dis...
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.