Is it permissible to perform the Sunnah rawatib prayer—the one before the obligatory prayer—in the mosque, and the one after—in the home?

There is no harm in that, though performing supererogatory prayers at home is better, based on the Prophet's (peace and blessings be upon him) saying: "Perform prayers in your homes, for the best prayer of a person is in his home, except for the obligatory prayer." Al-Nawawi explicitly stated the desirability of performing the regular supererogatory prayers at home, and this is the view of the majority, affirming that this opinion has no opposing view.