(א) מֵאֵימָתַי מַזְכִּירִין גְּבוּרוֹת גְּשָׁמִים. רַבִּי אֱלִיעֶזֶר אוֹמֵר, מִיּוֹם טוֹב הָרִאשׁוֹן שֶׁל חָג. רַבִּי יְהוֹשֻׁעַ אוֹמֵר, מִיּוֹם טוֹב הָאַחֲרוֹן שֶׁל חָג. אָמַר לוֹ רַבִּי יְהוֹשֻׁעַ. הוֹאִיל וְאֵין הַגְּשָׁמִים אֶלָּא סִימַן קְלָלָה בֶּחָג, לָמָּה מַזְכִּיר. אָמַר לוֹ רַבִּי אֱלִיעֶזֶר, אַף אֲנִי לֹא אָמַרְתִּי לִשְׁאוֹל, אֶלָּא לְהַזְכִּיר מַשִּׁיב הָרוּחַ וּמוֹרִיד הַגֶּשֶׁם בְּעוֹנָתוֹ. אָמַר לוֹ, אִם כֵּן, לְעוֹלָם יְהֵא מַזְכִּיר:
(1) From when do we begin to mention the power of [bringing] rain [in the Shemoneh Esrai], Rabbi Eliezer says, from the first day of the festival [Sukkot]. Rabbi Yehoshua says, from the last day of the festival [Sukkot]. Rabbi Yehoshua said to him [Rabbi Eliezer], since rain on the festival is nothing but a sign of a curse why should he mention it? Rabbi Eliezer said to him: I also did not say to request [rain] but to make mention, He causes the wind to blow and the rain to fall in its due season. He replied to him: if so one should at all times make mention of it.
Kochav, you are missing the answer to the last question about the second mishna, as well as summary points about the first one. Please revise in order to receive full credit. 5/6
מאימתי מזכירין גבורות גשמים- From when do we mention G-d's power through rain?
הואיל ואין הגשמים אלא סימן קללה בחג- since rain is nothing but an unfavorable sign on a festival
Summary of the mishna:
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Rough Translation:
From when do we mention G-d's power through rain? Rabbi Eliezer says that from today first day of the chag is good. Rabbi Yohshua says the last day of the holiday is good. Rabbi Yehoshua says to him. Since rain is nothing but an unfavorable sign on a festival why should we remember it? I am not asking for rain I am just saying it is a good time to mention it the rest of the year.
(ד) הִגִּיעַ שִׁבְעָה עָשָׂר בְּמַרְחֶשְׁוָן וְלֹא יָרְדוּ גְשָׁמִים, הִתְחִילוּ הַיְחִידִים מִתְעַנִּין שָׁלשׁ תַּעֲנִיּוֹת. אוֹכְלִין וְשׁוֹתִין מִשֶּׁחֲשֵׁכָה, וּמֻתָּרִין בִּמְלָאכָה וּבִרְחִיצָה וּבְסִיכָה וּבִנְעִילַת הַסַּנְדָּל וּבְתַשְׁמִישׁ הַמִּטָּה:
(4) If the seventeenth of Mar-Cheshvan has come without the rain having yet descended, private individuals commence to keep three fast-days. On the night beforehand, they are allowed to eat and drink, and it is allowed [on the fast day itself] to work, to bathe, to anoint the body, to wear [leather] shoes, and to perform the marriage duty.
1. What do we do if rain has not fallen by the seventeenth of Mar-Cheshvan?
If there is no rain fast for three days
2. What are we allowed to do, and what aren't we allowed to do?
At certain times you can where leather shoes, eat, drink, take a bath, work, to anoint the body, and to perform the marriage duty.
3. What holiday(s) is this similar to? Why?
I think Shmini Atzeret is a bit similar to Yom Kipur and other fast days because if there is no rain, you have to fast for three days
If we only ask for rain at certain times of the year, why do you think we mention it twice every day as part of the Shema?
I think we mention it because we need rain for the food everyday.