School started in September, atround the end of summer.

Sounds more accurate/natural for September to say "around the end of summer".

It finishedended back in June, in the early summer.

This sounds a bit more natural, but nothing wrong with what you wrote.

It was really hot and the sun was warmly shining warmly.

School mostly starts and finishes on same date.

There weren’t a lot of interesting things which happened last year, but I got closer to my classmates.

"Not many interesting things happened last year" might sound more natural, but this is fine too.

I had Math, English, Croatian, PE, Art, Science, and History…. I usually had six or seven lessons.

I didn't havechoose any favourite subjects because I liked them all.

Just saying "I didn't have any favourite subjects" is also fine.

I felt great and was excited whole year.

While your sentence wasn't grammatically wrong, it sounded a bit unnatural.

School started in September, at the end of the summer.

It finished (or it ended) in June, in the early summer.

It was really hot and the sun was warmly and shining. (and the warm sun was shining)

School mostusually starts and finishes on same date.

I didn't have favourite subjects because I liked them all.

I felt great and excited the whole year.