Is he who doubts the disbelief of Christians and Jews, or does not declare them disbelievers, himself considered a disbeliever? And how can one overcome these intrusive thoughts while believing that Islam is the truth?

There is no harm in dealing with disbelievers. The Prophet, peace and blessings be upon him, used to deal with them. It is necessary to differentiate between dealing with them and believing in their disbelief or hating them because of their disbelief. If you have intrusive thoughts (waswās) regarding your doubt about their disbelief, do not pay attention to them and repel them. Seek refuge with Allah from the accursed Satan. As long as the matter does not exceed being a mere intrusive thought, it does no harm.