I'm not good at explaining grammar but I think I can at least tell you which particles I would use if I were to say your sentences:
1.春にはorはたくさん雨が降る。
... which is like "In spring/As for spring, we have a lot of rain," since neither 春に~~ or 春で~~ sounds natural here, although you would use に when you say:
京都では、春にたくさん雨が降る。In Kyoto, it rains a lot in spring.
春にたくさん雨が降るのは、~~~だからだ。It is because ~~~ that it rains a lot in spring.
春にたくさん雨が降ると、夏は~~~。When it rains a lot in spring, ~~~ in summer.
(The に is often left out in daily conversation.)
2.[春雨]{はるさめ}だ。
... which means "It's spring rain", since none of 春に雨だ, 春で雨だ, or [春雨]{はるあめ}だ sounds right to me.
3.雨の日はきれいだ。
... which means "Rainy days are beautiful." or "(Something) is beautiful on rainy days." Neither 雨の日にきれいだ。 or 雨の日できれいだ。 sounds right. You could use 雨の日に in:
事故は、雨の日に起こりやすい。Accidents are more likely to occur on rainy days.
雨の日に学校に行くのは、いやだ。I don't want to go to school on a rainy day.
4.今月の三十一日に日本に行く。
Basically you can use に for a specific point in time, eg: in/on/at + year/month/day/time etc. → ~年/月/日/時に. (In daily conversation you'd often leave out the に, as in 今月の31日、日本に行く。)
5.休みに日本に行く。
in/during + break; on + holiday etc. → 休みに
6.一時間で食べた。
in/within + period of time → ~~で. eg:
2週間でひらがなを[覚]{おぼ}えた。I learned hiragana in two weeks.
1年で100万円[貯]{た}めた。I saved ¥1,000,000 in one year.
7.日本語でおねがいします。
... which is like "In Japanese, please." You'd use を when asking for / ordering something, like コーヒーをふたつ、おねがいします。/ Two coffee, please. (The を can be left out in conversation, as in コーヒーふたつ、おねがいします。)