In your first sentence, either rainy or raining could fit, depending on what you actually want to say; "... because it is raining" indicates that water is physically falling from the sky right now, while "because it is rainy" indicates that it is the sort of day where rain is extremely likely to happen, but doesn't necessarily mean that rain is ...
To talk about the weather, we idiomatically use "it". It's raining (now) Yesterday it was raining all day. Yesterday it rained (at least once) To talk about the type of weather you might use "rainy". It is rainy in Wales (usually) Yesterday, it was rainy. It was a rainy day. Note "I didn't go outside of house" is very non-idiomatic. Use "I didn't leave my home", for example.
The reason is that in the first sentence, "today is rainy", today is the object being described directly, so you don't need the pronoun 'it'. In the second however, there is a comma so after the comma, the 'it' pronoun is needed to make the sentence correct (hence the 'it's').
is it correct to say "today is rainy" or it is "today, it's rainy"?
What are the best foods to have on a rainy day? What are the best foods to have for a rainy day? Which preposition would be appropriate here?
grammar - ON a rainy day or FOR a rainy day? - English Language ...
Today is good. "Rainy" and "good" are both adjectives, so: Today is a rainy day. Today is rainy. But "frost" is a noun, so to make parallel sentences, you would have to use the adjective, "frosty": Tomorrow will be a frosty day. Tomorrow will be frosty. Alternatively, if you want to use the noun "frost", you could say, "Tomorrow there will be ...