Is the regular Sunnah of the Ishā' prayer (two rakahs after it) counted as part of the Tarāwīh prayer, or is it separate from it, especially since the Imam begins the Tarāwīh immediately after finishing the Ishā' prayer?

The regular Sunnah prayer after Isha is not part of the Tarawih prayer. Rather, it is performed before commencing the Tarawih prayer. It is permissible to perform it after Tarawih, but it is preferable to perform it before Tarawih.