Hello, I'm aAmerican and i'I’m sixteen years old

I have assumed you meant ‘American’ here, but if you mean your name is America, the correction is still the same - capital A! And a capital ‘I’ too.

I like listening to music and eating

The two verbs should match; in this case, you wrote ‘listening’, so ‘eating’ makes sense (i.e. both ‘-ing’)

My favorite colors is Gare green, black, and white

‘Are’ not ‘is’, as you’ve mentioned more than one colour. There’s also no need for a capital letter for colours. And I added an Oxford comma (the last comma in a list), which is American style (since you used the American spellings ‘favorite’ and ‘colors’!).

I like to sleep and I; I usually sleep six or eight hours

A semicolon works here, as the part after it is related to the first bit of your sentence (‘I like to sleep’). I added ‘usually’ to make things sound a bit more complete and natural!

I have a cat, but I don't have dogs

A comma is good here, as each part (‘I have a cat’ and ‘I don’t have dogs’) could stand on its own, each as a sentence.

Hello, I'm aAmerican and iI'm sixteen years old.

The first letter of "America " and "American" and "I" are always capitalized. We always end a sentence with a period (.)

I like listening to music and eating.

Parallel construction: I like listening, I like eating.

My favorite colors is Gare green, black and white.

"Colors" is plural so it takes the plural verb "are" in this sentence.

I love to eat chicken.

I like to sleep andso I sleep for six or eight hours.

I have a cat but I don't have any dogs.

Hello I'm aAmerica and iI'm sixteen years old

Names start with a capital letter
We also capitalise "i" or "i'm" so it is "I" or "I'm"

I like listening to music and eating

My favorite colors is Gare green, black and white

You use "are" as there is more than one colour

I like to sleep and I sleep six or eight hours (every night)

I have a cat but I don't have a dogs