Is it permissible to recite the first Tashahhud in a supererogatory prayer and then say the Taslim, or must both Tashahhuds be recited together? And is it valid to vocalize the recitation in a supererogatory prayer from after Maghrib until the time of Fajr?

Your question about performing supererogatory prayers (nawafil) with four continuous rak'ahs is permissible according to the majority of scholars, though it is preferable to limit them to two rak'ahs. As for sending prayers upon the Prophet after the tashahhud in a supererogatory prayer, it is obligatory according to many scholars, and it is not permissible to say the taslim before it; otherwise, the prayer becomes invalid according to the Shafi'is and Hanbalis. As for the regular sunnah prayers (rawatib), it is sunnah to recite silently. The same applies to supererogatory daytime prayers. However, for supererogatory nighttime prayers, it is sunnah to be moderate between audible and silent recitation, and it is also said that audible recitation is sunnah or one is given a choice.