Is there a special intention for Tarawih prayer, or just the intention of performing [prayer in general]? And for the Friday Sunnah prayer at home, should the intention be for Dhuhr or for Jumu'ah?

When commencing the Tarawih prayer, one should specify its intention, as it is a specific supererogatory prayer (nafl mu'ayyan). This is what Ibn Qudamah mentioned in "Al-Mughni."

Although some jurists, such as the Hanafis and Malikis, hold that the intention for any absolute supererogatory prayer suffices for the Tarawih prayer, it is more cautious to specify its intention, based on what the author of "Al-Mughni" stated.

As for the Friday sunnah prayer (ratibah) performed at home after the Friday prayer, it is the sunnah of Friday and is not performed as the sunnah of Dhuhr.