How can one get rid of intrusive thoughts about the the existence of Allah, the Qur'an, and Islam, and is it permissible for him to marry a Muslim woman while in this state?

One should turn away from satanic whisperings (waswas) and preoccupy oneself with dhikr (remembrance of Allah) and reciting the Quran, to prevent Satan from gaining control, and ask Allah to remove the affliction.

Waswas alone does not expel one from the fold of Islam, unless it is accompanied by heinous acts or thoughts of disbelief. What is uttered unintentionally does not lead to disbelief, for the pen is lifted from one afflicted with depression or obsessive-compulsive disorder (OCD) when their intellect is impaired.

Obsessive-compulsive disorder is not among the defects that must be disclosed before marriage, unless health and safety were stipulated as a condition.