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Ethics
Knowledge

Is disregarding normative claims (epistemic like "you ought to believe true things" or moral like "you ought to donate to Oxfam") with full knowledge and understanding of them irrational? Is rationality grounded in other way than that we just all seem to participate in "rationality project" from the get-go (question of "why be rational" seems to be self-defeating as we look for reasons to be rational..)?

For the first question, let's focus on the practical claim. Suppose you judge that, all things considered, you ought to donate to Oxfam today but you experience no motivation to do so. You never form the intention to donate or form a desire to do so. A version of what is called "motivational judgement internalism" holds that this psychology would constitute a rational failing. This version maintains that it is a rational requirement that whenever someone judges that they ought to A, they experience at least some motivation to A. I won't take a stand on whether this thesis is true, but this should give you some terminology to search for. Your second question asks what explains rational requirements and why we should follow them. One strategy is to try to show that all rationality just is instrumental rationality. Since it seems unproblematic why (all else equal) we should take the necessary means to our ends, showing that all rational requirements amount to this would be progress. How would this...
Knowledge

A typical response to global skepticism (skepticism which claims that we can know nothing), is that such a position is self-defeating. However, couldn't the global skeptic respond by stating that such an objection relies on the objector having knowledge of the truth of the law of non-contradiction, which the skeptic claims we don't know? Thus, the skeptic could argue that they know nothing can be known while having the privilege of also incoherently claiming that they still know certain things. Would it even be possible to even intelligibly respond to such a claim, since it seems when one begins to reject things like the laws of logic, then intelligible discourse becomes impossible and were left with assumptions?

Fascinating questions. The view that no one can know anything does not seem self-defeating to me. You think this version of global skepticism defeats itself because it implies that the view itself is unknowable. But why does this defeat the view? Even if we cannot know the view to be true, there still may be sufficient reasons to believe it. Nevertheless, suppose you are a global skeptic who wishes to maintain that you know the truth of global skepticism. You believe global skepticism and its implication that you do not know the truth of global skepticism. Does the fact that you do not know the law of non-contradiction give you a reason to affirm that you also know global skepticism is true? I do not see why. To be consistent in your global skepticism, you should admit that you don't know whether there are any true contradictions. But that doesn't give you a positive reason to believe that when you say "I know no one can know anything," it is both true and false.