My name is Peter and I haveam 20 years old.

We use the verb to be, instead of to have

I am student in a university.

you can say: "I am a university student"

I've lived in Austria since 2 years ago.

past continuous

or
"I've lived in Austria for 2 years."

I like listening to music and watching movies.

I have one brother and two cats.

plural

My English is not perfect but I will try to learn every days.

future tense

My name is Peter and I haveam 20 years old.

I changed "have" to "am".

I am student in a university.

You could also say I am a university student, but your writing is grammatically correct!

I live in Austria since 2 years.

I have lived in Austria for 2 years. Or, if you want to use "since", you could say: Since 2023, I have lived in Austria.

I like listen music and watching movie.

Option 1: I like to listen to music and watch movies.
Option 2: I like listening to music and watching movies.

I have one brother and two cat.

I have one brother and two cats. In English you have to match the adjective to the noun in terms of plural/singular.

My English is not perfect but I try learn every days.

My English is not perfect, but I try to learn every day. (You're doing great!).

My name is Peter, and I haveam 20 years old.

I am a student in a university.

"in a university" is not wrong, but this feels more natural

I have lived in Austria sincefor 2 years.

I like listening to music and watching movies.

I have one brother and two cats.

My English is not perfect but I try to learn every days.

My name is Peter and I haveam 20 years old.

"I am X years old" is correct.

I have lived in Austria sincefor 2 years.

I like listening to music and watching movies.

listen music -> listening to music
watching movie -> watching movies

I have one brother and two cats.

cat -> cats: plural

My English is not perfect but I try to learn every days.

try learn every days -> try to learn everyday.