Simchas Torah
Yesod VeShoresh HaOvadah (ch.16) writes, "Whoever is happy with the Torah on this day, he is guaranteed that his children and descendants will always be Torah scholars."
Shlamei Chagigah (p.292: printed year תק"ג) writes, "Fortunate is the person who rejoices with the Torah on Simchas Torah, for this will rectify his soul for the times that he disgraced the Torah. Additionally, there's a guarantee from early scholars that whoever is happy with the Torah on this day will have offspring who will always study the Torah. There was a Rav who was very happy on Simchas Torah, and his descendants were renowned scholars for three consecutive generations. Everyone knew it was because of his happiness with the Torah on Simchas Torah. He rejoiced and danced with the Torah in a way that wasn't done by other rabbanim and gedolei Yisrael. Despite his prestigious Rabbinic position, he would jump and dance before the Torah and encourage others to do the same." In this merit, he merited children and grandchildren, exceptional talmidei chachamim.
The Shaar HaKavanos writes, "My teacher (the Arizal) went around the bimah... danced and sang with all his strength. Then, on motzei Yom Tov, he went to several batei midrashim to dance with the Torah (called in Eretz Yisrael הקפה שניות), and this lasted many hours."
Maaseh Rav states that the Vilna Gaon zt'l "would go before the sefer Torah with immense joy, his face was like a burning fire, and he would clap and dance with all his might before the Torah."
The Satmar Rebbe zt'l danced with all his strength on Simchas Torah, although he was weak. When people asked him about this, he said, "Chazal (Succah 48.) say, שמיני עצרת רגל בפני עצמה היא.40" On Shemini Atzeres, everyone receives another רגל (foot) to help him dance.
Dancing with the Torah on Simchas Torah is mesugal for bearing children. The Yismach Yisrael zt'l learns this from Michal, who was upset with her husband Dovid Hamelech when she saw him dance with all his might before the aron hakodesh (see Shmuel 2, 16:23). She was punished that she couldn't bear a child. We understand from this that when one does the opposite and is happy with the Torah, he will be rewarded with children.41
40 Literally, this means Shemini Atzeres is its own holiday, and not a continuation and final day of Succos.
41 It is brought down in Shivchei HaBaal Shem that one Simchas Torah, the students of the Baal Shem Tov zy'a were dancing joyously, and one of the poorer students left the holy dancing and sat on the side, disappointed. He couldn't continue dancing because his shoe had ripped. Udel, the Baal Shem Tov's daughter, noticed what happened and told the student, "If you will guarantee me that I will bear a son this year, I will get you a pair of new shoes." (She had shoes in her store.) The student gave his word. She immediately brought him the shoes. That year, she gave birth to the great tzaddik, Rebbe Boruch'l of Mezhibuzh.
A member of our chaburah didn't have children. One Simchas Torah, he danced with all his might with the Torah, and nine months later, he had his first child. The following Simchas Torah, he did the same, and once again, nine months later, he had his second child. He had a friend who also didn't have children. When he heard that his friend had two children in the merit of dancing with the Torah, he wanted to do the same. However, he was embarrassed to dance with all his might in the beis medresh where he usually davens, so he went to an old-age home where no one recognized him, and he danced enthusiastically and honored the Torah immensely. Nine months later, his first child was born.
Rebbe Yehoshua of Belz zt'l said in his father's name, Rebbe Shalom of Belz zt'l, "I can't tell you what the hakafos are about. But I can tell you this: All tefillos that didn't go up the entire year, go up during the hakafos." This is hinted in the words (see Succah 48a) פז"ר קש"ב which can mean פז"ר the tefillos that were scattered and didn't go up; קש"ב on Shemini Atzeres, Hashem listens to those tefillos.
The Minchas Elazar zt'l (Shaar Yissaschar, Zman Simchoseinu 17) writes that the Baal Shem Tov zt'l said, quoting Eliyahu HaNavi, that whoever davens with kavanah on Hoshanah Rabba, Shemini Atzeres, and Simchas Torah, he will merit to have kavanah the entire year.
The Beis Aharon zt'l (Pesach p.94.) writes, "Dancing on Yom Tov is the highest form of tefillah... for it is a prayer with כל עצמותי all one's limbs."
The Ateres Yehoshua (ד"ה במאמרם) explains the Mishnah (Negaim 12:1) בית עגול אינו מטמא בנגעים, that due to the עגולים, our dancing in a circle on Simchas Torah, אינו מטמא בנגעים all plagues and troubles go away.
The Chida teaches that by dancing on Simchas Torah, we knock down the iron walls that separate us from Hashem.
On Simchas Torah, we read the Parasha of וזאת הברכה, the brachos Moshe Rabbeinu gave Bnei Yisrael. As we read them, we receive those brachos again. Rebbe Henoch of Alexander zt'l said this is why birchas kohanim isn't performed on Simchas Torah. We are receiving Moshe Rabbeinu's brachos, so we don't need any other brachos.