Is a father's joking address to his daughter as "O Noah's Ark" considered mockery that necessitates an act of disbelief (takfir), and what is the ruling concerning a person afflicted with obsessive doubts (waswasi) who hesitates in declaring a specific individual to be a disbeliever?

There is no harm in calling children by names other than their own out of affection. However, calling a daughter "O Noah's Ark" is not good because the ark is one of the شعائر الله (symbols/rituals of Allah), and this meaning is not realized in a daughter.

Nevertheless, the father does not commit an act of disbelief by doing so. Speech that is susceptible to both disbelief and non-disbelief does not lead to disbelief unless the speaker intends disbelief. If there is even one way for a Muslim to remain in their Islam, it should be acted upon.

It is not permissible for someone afflicted with obsessive doubts (al-waswas) to accuse a specific individual of disbelief, because such a person is not sound in their judgments. The prescribed approach is to avoid declarations of disbelief. We ask Allah to cure the one suffering from obsessive doubts of their affliction, and we advise them to supplicate and consult a psychiatrist.