What should one do who is assailed by doubts about the existence of past nations and the prophethood of Muhammad (peace be upon him), and about Paradise and Hell, despite his attempts to intellectually and jurisprudentially refute these doubts, and his perseverance in acts of worship, even while suffering from psychological deterioration due to these thoughts? And is this considered disbelief?

This situation is a case of waswasah (whisperings/obsessive thoughts), and Allah has overlooked them as long as they are not acted upon or spoken. Thoughts and self-talk, if not settled upon and believed by their owner, are pardoned by consensus. The servant's dislike of these waswasah is evidence of the soundness of their faith, as they only afflict those whose faith is complete. This person belongs to this category, as described in the question, where they refute these doubts intellectually and religiously and seek refuge from Satan. Doubting the existence of previous nations or of Prophet Muhammad, peace be upon him, is outside the scope of sound reason and logical thinking, because continuous transmission (al-naql al-mutawatir) necessitates belief.