Shmuel Hakatan instituted the bracha against heretics in Shemone Esrei. He once was leading davening and forgot the text of that bracha and ws delayed 3 hours until he was able to recall the language. The Gemara challenges this by a rule that if someone is not saying the bracha against heretics, we suspect they are a heretic and we remove them from the "amud." The Gemara answers with a dispute between Abaye and Rava. According to Abaye, people who were always righteous simply do not turn wicked and that's why Shmuel Hakatan was not suspected. Rava holds even fully righteous people can turn bad, and he suggests that the only reason Shmuel Hakatan was not removed was bc he had started the bracha and was therefore clearly not a heretic. (the dispute between abaye and rave is regarding the identity of Yannai Hamelch and Yochanan Kohen Gadol)

7 brachos on Shabbos correspond to the 7 "kolos" in Tehillim 29. the 9 brachos on Rosh Hashana correspond to the 9 names of Hashem in Chana's prayer, bc she was answered on Rosh Hashana. The 24 brachos on fast days correspond to the 24 languages of Tefillah sung by Shlomo when the Aron was blocked from entering the Kodesh Kodshim.

We learned in the Mishan that R' Yehoshua allows for an abridged version of Shmone Esrei. There are two opinions of how it is shortened. Either each bracha is shortened but there are still 18 brachos. Another option is called "Havineinu" where the middle section of Shmone Esrei is consolidated into one large bracha with each bracha having just one phase that relates to its theme.

One can't say Haveineu option of Motzei Shabbos bc there is not bracha where one can "insert " the havdalah.

One can't say Havineienu option during the winter bc there is no where to place "v'seen tal umatar." (it is confusing to add it just that phrase in the middle of Havineinu.)

If one forgot to say V'sein Tal Umatar and remembered before Shome Tefillah, they can insert it there. If they already passed Shome Tefillah, they have to go all the way back.

If someone forgot Yaaleg Vyavo- anytime before they finished Shmone Esrei they go back to go back to beginning of R'tzei. There are two versions in the Gemara. The lenient version says that if someone didn't take three steps back or even if they did but they still have supplications to say, they can still go back to rtzei and not repeat the whole TEfillah. The second stringent opinion says one can only be lenient if they haven't yet taken three steps back AND they still have supplications to say.