Is doubting God, the Quran, and the Day of Judgment, while simultaneously believing in God, considered disbelief? What are the ways to overcome such doubt? And what are the proofs of God's existence to confront it?

What you are suffering from is a whisper (waswas) that does not harm the believer as long as he dislikes it. Therefore, you must fight against it and seek refuge in Allah from Satan when it occurs, and say: "I believe in Allah," and strive in supplication. You will be rewarded for this struggle. As for the proofs of Allah's existence, every atom in the universe bears witness to His creation, and His signs in the horizons and within yourselves are the greatest witness to His existence, wisdom, power, and mercy. It is recommended to read the book "Aqeedah Al-Mu'min" (The Creed of the Believer) by Sheikh Abu Bakr Al-Jazairi, and the book "Al-Aqeedah fi Allah" (The Creed in Allah) by Sheikh Omar Al-Ashqar.