Q: as i remember: "Raining cats". とはどういう意味ですか?
A: "Raining cats and dogs." It means it's raining heavily. But people don't actually say this, it's a weird and uncommon phrase.
Q: Raining cats and dogs とはどういう意味ですか?
A: something that you say when it is raining heavily
Q: Raining cats and dogs.
A: It's an expression which means that there's heavy rain.


Q: 1. I'm depressed when it's raining.
2. Raining makes me depressed. この表現は自然ですか?
A: 1. I get depressed when it rains.
2. Rain makes me depressed.
Q: Raining. Want sleep. But I slept too much today. この表現は自然ですか?
Sentences have to have three words.
If you wanted to you could probably say:
It is raining. I want to sleep. But I slept too much today.
Q: Raining is normal in this season in Japan because it's rainy season. この表現は自然ですか?
A: Rain is normal during this season in Japan because it's the rainy season.

"Rain" (verb) already has a noun, in fact it's the same word, so you don't need to use 動名詞.

"In this season" is usually ok, but using "in" twice is a bit strange.

There is only one rainy season, so you need "the". The other seasons (Summer, etc.) are proper nouns, so they don't need it.
Q: Raining day~
A: I would say: Rainy day.~
Or: It's raining today.~