When should a person who is davening at home daven Musaf on Shemini Atzeres?

As we have learned, Mashiv Haruach is not said until an announcement is made in shul. A person who is davening at home must also wait until that announcement is made. If there are several shuls in the city, it is plausible to say that one should wait until the announcement is made in the shul where he regularly davens. A person who lives in a village without a minyan should wait until the end of the morning (before chatzos) to say Musaf with Mashiv Haruach. At that point, he can assume that the tzibur has already said Mashiv Haruach. If he is certain that they have not yet davened after seven hours of the day have passed, some say that he should continue waiting until they daven while others say that he should daven on his own and omit Mashiv Haruach.

סי' קי"ד ס"ק ז'-ט' ביאורים ומוספים דרשו 13,14 

When should a person who is davening at home daven Musaf on the first day of Pesach?

The Gemara teaches that tal (dew) will never be held back from mankind. However, the dew will not necessarily be tal shel brochah (dew which brings blessing). Since tal will never be withheld, all opinions agree that if one did not mention tal in the summer, he does not have to go back. The minhag Sefard, however, is to mention it l’chatchila so that it will be directed for blessing. The minhag Ashkenaz is to only incorporate it as a request in Birchas Hashanim. A follower of the minhag Ashkenaz who is davening by himself on the first day of Pesach should daven Musaf as early as possible (i.e. before the chazan stops saying Mashiv Haruach) and say Mashiv Haruach. In contrast, a person who keeps minhag Sefard should daven Musaf later than the tzibur so he can say Morid Hatal after the chazan has already said it.

ס"ב ס"ק י"א-ט"ז ביאורים ומוספים דרשו 17

When is it necessary to go back if one erroneously said Morid Hageshem?

Geshem in the wrong season is not beneficial for the world. Therefore, mentioning it at the wrong time can affect the validity of one’s Shemoneh Esrei. The following are some pertinent halachos: 1) If one immediately corrects himself, the poskim disagree about whether he may continue onward. Some say that since mentioning geshem is a klalah, an immediate correction is ineffective and he must go back to the beginning of the brocha. 2) If one mistakenly mentions geshem without correcting himself immediately, he must go back to the beginning of the brocha. B’dieved, if he only went back to the words rav l’hoshia he is also yotzei. 3) If one only realized his mistake after finishing the brocha, he must begin Shemoneh Esrei again. Some poskim say that if he realized his mistake immediately after saying Hashem’s name at the end of the brocha, he should say the words lamdeini chukecha and return to the beginning of the brocha.  

ס"ד ס"ק י"ח-כ' ביאורים ומוספים דרשו 18

• The poskim disagree on whether the word geshem should be enunciated with a segol beneath the gimel (geshem) or whether it should be pronounced with a kamatz (gashem).

• In communities where Tefilas Geshem is said before the quiet Shemoneh Esrei, it serves as the announcement to begin Mashiv Haruach.

• Just as Mashiv Haruach cannot begin without an announcement, it cannot end without an announcement.

• What is the chiddush of R’ Akiva Eiger regarding one who omits Mashiv Haruach on Friday night?

• Can Mashiv Haruach be added at any point in the brocha?

• Can an omission be rectified after the brocha is completed?