What should I do to be convinced of God's existence, and to know Islam as the Messenger, peace and blessings be upon him, defined it for the early Muslims, not merely as rulings, lawful and unlawful? And what is the ruling of Islamic law if I die while doubting God's existence? And is my destiny Hellfire?

The doubts you mentioned about the existence of God are mere whisperings [waswas], having no reality, and they do not affect the validity of your faith as long as you dislike them and are repelled by them. So, strive against them, and you will be rewarded for that. If you die in this state, you are upon goodness. The proofs of God's existence and His Oneness are very numerous; every moving and still, wet and dry thing bears witness to that. We advise you to review the book "Al-Aqeedah fi Allah" (The Creed Concerning God) by Dr. Omar Al-Ashqar to strengthen your faith.