Is doubting the religion and its rulings, and questioning the authenticity of the Quran and prayer, considered apostasy, or is it a form of intrusive thoughts (Waswas)? And how can one deal with this doubt?

You are afflicted with waswasah (whisperings/obsessive thoughts), which is a dangerous disease that corrupts both religious and worldly matters. You must close this door and not give in to it. Rather, strive to repel it and turn away from it. If any of it comes to you, stop thinking about it and seek refuge with Allah from Satan. These whisperings do not affect the validity of your faith, and you are upon good as long as you dislike them, and you will be rewarded for repelling them. The cure for waswasah in purity, prayer, and other matters is to turn away from it and not give it any attention.