Is it sufficient to recite only Surat Al-Fatihah in the Sunnah prayer? And did the Messenger of Allah, peace and blessings be upon him, perform it upon entering the mosque or after the obligatory prayer? And is a single Tasbeeh (glorification) sufficient in bowing and prostrating for the obligatory prayer?

There is no Hadith (definition) that limits the Prophet (peace be upon him) to only recite Al-Fatihah in the Sunnah (definition) prayers. However, he would sometimes lighten some Sunnah prayers by reciting Al-Fatihah and a few verses. The confirmatory Sunnah indicates the permissibility of limiting oneself to Al-Fatihah in prayer generally, as in the Hadith of Mu'adh. As for the tasbih (glorification) in bowing (ruku') and prostration (sujud (definition)), it is recommended according to the majority of scholars, and obligatory according to the Hanbalis, with one recitation being sufficient for the obligation.