How can one deal with doubts about the truthfulness of the Prophet Muhammad (peace be upon him) and the Noble Qur'an, which arise from unanswered supplications or unfulfilled reliance on God in some situations? And is the aforementioned doubt considered disbelief (kufr)?

If the obsessive thoughts are merely satanic whisperings that you detest, then they will not harm you nor affect the soundness of your faith. You must ignore them. These obsessive thoughts can be dispelled by recalling that the forms of answering prayers vary, and it is not necessary for the servant to receive exactly what they prayed for, and that Allah is more merciful to His servant, and that any calamity that befalls them is due to their sins, and that the servant has limited intellect and does not know where their true benefit lies. As for what you mentioned about Allah's tidings not coming to pass, know that all of Allah's tidings are truthful, and it is impossible for Him to announce something that does not occur as He announced it. Scholars can clarify this issue, and you should refer to them and be careful not to slacken in your struggle against these obsessive thoughts.