Is a person considered an infidel if he mistakenly lies about the Prophet (peace be upon him) and then retracts it, or if he says "I am an infidel" in jest, or if he considers someone an infidel in his heart for doing a certain thing?

Lying about the Prophet (peace be upon him) is forbidden, but it is not disbelief (kufr). One who errs in narrating a hadith unintentionally does not incur sin. Declaring a Muslim a disbeliever is interpreted as minor disbelief, based on the hadith narrated by Thabit ibn ad-Dahhak, "And whoever accuses a believer of disbelief, it is as if he has killed him." One who declares a Muslim a disbeliever does not himself become a disbeliever; rather, the accused remains his Muslim brother.

Invoking Allah's name while committing a sin is forbidden, but it does not expel one from the fold of Islam. Reciting the Basmala (Bismillah) before committing an inherently forbidden act (such as adultery or drinking alcohol) is prohibited. A Muslim declaring himself a disbeliever expels him from the fold of Islam, unless he is under duress.