Is it obligatory to perform the Sunnah prayers (Rawatib) – those before and after the prescribed prayers – after the time for the prayer has begun, or can they be performed before it, especially since the Prophet (peace and blessings of Allah be upon him) used to perform them at home and then go out for the prayer?

The pre-Salah regular Sunnah prayers are not performed except after the time for the prayer has begun. Your inference from the action of the Prophet, peace and blessings be upon him, is incorrect, because he used to perform them after theWaqt (time) had begun, and then he would wait until the Iqamah (call to commence prayer) was made.