정신적으로 피곤하다

1. 저는 공부했습니다.

It is not wrong to say this, but I think it would be better if you specify what you studied, if that's Korean that you spent time studying,

you can add "한국어를" as an object,

so the whole sentence would go like this,

"저는(나는) 한국어를 공부했습니다."

"저는(나는) 한국어를 공부했어요."

2. 너무 많이 배웠습니다!

많이 배웠습니다!

The difference between the two is the use of "너무", and I feel like when it comes to learning there is never too much, so I would use the latter one, which is "많이 배웠습니다", but if you want to say, I feel like I bit more than I could chew, then it's alright to say "너무", and it would mean I learnt so much that I reached a point where it is almost impossible to absorb more.

3. "깊이 자고 싶습니다." is okay,

 but what could sound more natural is like the sentence below, and it means the same.

"잠을 푹 자고 싶습니다."

Hope this helps, Gwen. :)