The first sentence is correct.

The second sentence should be "Yesterday I wore a shirt because the weather was hot"

"I wear shirts in (the) hot weather" <-- (the) is optional. 

Today I am wearing a shirt because it is hot. (correct) You don't wear a shirt every day so you can't say  Today I wear.....

You should say  Today I am wearing... because it's happening today. The present continuos tense is used for showing the action that is happening right now, at the moment) and even today because it's a period when you don't do what you used to do but smth different.