Is it necessary to perform the marriage contract (nikah) in the mosque? Is there an authentic hadith that obliges this? And where did the Prophet (peace be upon him) عقد the marriage of his daughters?

There is no authentic Hadith indicating the desirability of performing the marriage contract (nikah) in the mosque. Rather, some jurists deemed it desirable for blessings and for publicizing the marriage, while some based this on a weak Hadith. Furthermore, many people today prefer it to avoid religious transgressions. As for where the Prophet, peace and blessings be upon him, contracted the marriages of his daughters, there is no narration regarding that, and it appears it was done in ordinary gatherings or houses.