Is doubt disbelief, and does a person become a disbeliever by doubting verses and hadiths, and in the Qur’an and Sunnah, with the presence of continuous obsessive whispers, and is what the questioner feels considered disbelief?

The best cure for waswas (whisperings/obsessive thoughts) is to ignore them. Strive against your soul to cast them away and pay them no heed. Know that you will be rewarded for resisting them. As long as you dislike them and are repulsed by them, they will not harm you. What you are experiencing is not doubt, but rather waswas from Satan. So, whenever Satan whispers to you that you are doubtful, repel him and seek refuge in Allah from his evil. Say to him, by your very state: "Rather, I am certain of what Allah has revealed to His Prophet, peace and blessings be upon him."