In Arbit on the night of Shemini Aseret, we recite the chapter of Tehillim "La’menase’ah Al Ha’sheminit" in place of "Ke’ayal Ta’arog Al Afikeh Mayim." The word "Ha’sheminit" ("the eighth") alludes to Shemini Aseret, which is the eighth day when counting from the beginning of Succot, and thus this Psalm is relevant to this holiday. Furthermore, the second verse of this chapter prays, "Hoshi’a Hashem Ki Gamar Hasid Ki Fasu Emunim Mibeneh Adam" ("Save [us], Hashem, for the righteous one is gone; for there are no more faithful ones among people"). The Midrash says that this prayer was recited by Yehoshua after the death of Moshe Rabbenu, who was the "Ne’eman" – the most faithful servant of God. Yehoshua begged the Almighty to save Beneh Yisrael after Moshe’s passing, and this prayer is appropriate to recite on Shemini Aseret and Simhat Torah, when we read of Moshe’s passing (in Parashat Vezot Haberacha).
There are those who have the custom to stand for the recitation of Kaddish on the night of Simhat Torah. According to the view of the Arizal (Rav Yishak Luria of Safed, 1534-1572), however, this custom is incorrect. The Arizal understood that we stand for Barechu and Kaddish on Friday night because of the "Neshama Yetera" – the extra soul that we receive as Shabbat sets in. On Yom Tob, we do not receive a "Neshama Yetera," and therefore there is no reason to stand for Barechu or Kaddish on the night of Yom Tob.
In the Amida prayer on both days – Shemini Aseret and Simhat Torah – we refer to the holiday as "Yom Shemini Hag Aseret Ha’ze." If one mistakenly recited "Yom Hag Ha’Succot Ha’ze," he must repeat the Amida. Shemini Aseret is a separate holiday, and not the eighth day of Succot ("Shemini Regel Bi’fneh Asmo"), and therefore one’s prayer is invalid if he refers to the holiday as "Yom Hag Ha’Succot Ha’ze." If one realized his mistake before he completed the Amida, he returns to "Ata Behartanu." However, if he realized his mistake only after he recited "Yiheyu Le’rason Imreh Fi," he must recite the entire Amida again from the beginning. This applies to Kiddush, as well. If on the night of Shemini Aseret or Simhat Torah one mistakenly recited "Yom Hag Ha’Succot Ha’ze" in Kiddush instead of "Yom Shemini Hag Aseret Ha’ze" he must repeat the entire text of Kiddush.
For the same reason, we recite the Beracha of "Shehehiyanu" at Kiddush on both nights. Since Shemini Aseret/Simhat Torah constitutes a new holiday, which is not part of Succot, we recite "Shehehiyanu" over the occasion of this new holiday.
On the first night of the holiday, it is customary to wait until dark for the holiday to begin before reciting Kiddush. On the second night, too, one should wait until dark because otherwise he would be required to eat in the Succa, as we do on Shemini Aseret. Thus, on both nights Kiddush should be recited only after dark.
Summary: In Arbit on the night of Shemini Aseret, we recite the chapter of Tehillim "La’menase’ah Al Ha’sheminit" in place of "Ke’ayal Ta’arog Al Afikeh Mayim." In Kiddush and the Amida on both Shemini Aseret and Simhat Torah, we refer to the holiday as "Yom Shemini Hag Aseret Ha’ze," and one who mistakenly recites "Yom Hag Ha’Succot Ha’ze" must repeat the Kiddush or Amida. "Shehehiyanu" is recited at Kiddush on both nights. Kiddush should not be recited on either night before dark.