What are the connections between Sukkos and Yom Kippur?
The Aruch Hashulchan says after Hashem gave us the Torah even after we made the Egel Hazav(golden calf). This was on Yom Kippur the second Luchos where given to us this illustrated Hashem granting us forgiveness . Right away he gave the Jews a Mitzvah to make the Mishkan, which showed Hashem's love to Klal Yisroel because Hashem put his Shechinah amongst us. The Sukkah has the same message all year we do Aveiros comes Yom Kippur Hashem forgives us and then he tells us sit in the Sukkah and I will Protect you.The connection is that Yom Kippur and Sukkos are connected in that they illustrate Hashems forgiveness of our Aveiros as the Prototype of the Mishkan did Many generations earlier.
Here are a couple Midrashim that describe the connection between Rosh Hashanah, Yom Kippur, and Sukkot:
א"ר אבין משל לשנים שנכנסו אצל הדיין ולית אנן ידעין מאן הוא נוצח אלא מאן דנסב באיין בידיה אנן ידעין דהוא נצוחייא כך ישראל ואומות העולם באין ומקטרגים לפני הקב"ה בר"ה ולית אנן ידעין מאן נצח אלא במה שישראל יוצאין מלפני הקדוש ברוך הוא ולולביהן ואתרוגיהן בידן אנו יודעין דישראל אינון נצוחייא לפיכך משה מזהיר לישראל ואומר להם ולקחתם לכם ביום הראשון
R' Abin says: If two people have come before a judge, and we do not know who has been victorious, if one of them takes a palm branch in his hand, we know that he is the victor. So it is with B'nei Yisrael and the nations of the world: The latter come and bring accusations before Hashem on Rosh Hashanah and we do not know who has won. Since the B'nei Yisrael go forth from the presence of Hashem bearing their palm- ranches and their etrogim, we know that they are victorious.
“You shall take for yourself on the first day.” It is really the fifteenth day of the month, yet we are calling it the “first day?” R. Manna and R. Yehoshua said in the name of R. Levi: This is comparable to a country that is obligated to pay taxes to its king, and the king travels to collect his due. When he is 10 miles away, the most prestigious men of the country go out to praise him, and the king forgoes one third of the quota. When he is five miles away, the middle-class of the country go out and praise him, and the king forgives another third. When he enters the country, all citizens come out to greet him and praise him—men women and children— and the king forgoes the last third of the debt. He says: From this moment onward, we will begin a new accounting.
Similarly, on the eve of Rosh Hashana the great men of the generation fast and the Almighty forgives one third of their sins. Between Rosh Hashana and Yom Kippur, individuals fast, and G-d forgoes another third. On Yom Kippur everybody fasts, men, women, and children, and G-d says “What has passed is past; now let us begin a new accounting."
From Yom Kippur until Sukkos, all Jews are busy with mitzvos—this one with his Sukka, and this one with his Lulav. On the first day of Sukkos all Jews stand before G- d with their Lulav and Esrog for the sake of the mitzvah, and G-d says “What has passed is past; now let us begin a new accounting.”
Therefore Moses tells the people “And you shall take for yourselves on the first day...”
In General, the Chabad Chassidut emphasizes the view is that the whole time period from Rosh Chodesh Elul until after Simchat Torah is connected. Everything that is concealed on Rosh Hashanah and Yom Kippur is revealed on Sukkot and Simchat Torah. Also, It is specifically the Kabalat Hamalchut on Rosh Hashana and the Teshuva of Yom Kippur that brings us to the joy of Sukkot (Zman Simchatainu). See for example, here, here and here.
We say if the verdict of yom kippur was that we ever be exiled from our homes, may it be fulfilled with just this week in the sukkah.
(Hence an explanation of the Gemara -- sitting in the sukkah is like diluting wine; trying to turn G-d's harsh justice to mercy [by the above argument]. If it rains on your sukkah, the water gets thrown in your face -- nice try asking for mercy, no such sukkah-luck for you. If you need exile, you'll get real exile.)