She is training on the basketball court, and I am sitting beside it.

A comma is required punctuation to separate two independent clauses.

There are only five kids in the class today, because the winter holiday just began, and most kids are relaxing after final exams.

In this sentence, a comma before "because" is optional.
A comma before "and" is necessary because it separates two independent clauses.
Another way to explain this: if you put a period where I put the comma, and you delete the conjunction "and," you would have two complete sentences.

She looks very happy playing basketball with other kids,; I think this is exactly what I want.

These are two independent clauses without a conjunction (and, but, so, etc.). The only punctuation strong enough to connect them is a semicolon. Otherwise, you could use a period and make two sentences.

I see the class as an exercising and entertainment class but not a skill training class, lol.

I've brought my daughter to her basketball class at a college near our house.

I brought my daughter to her basketball class at a college near our house.

She is training on the basketball court and I am sitting beside it.

She was training on the basketball court and I was sitting beside it.

There are only five kids in the class today, because the winter holiday just began and most kids are relaxing after final exams.

There were only five kids in the class today, because the winter holiday just began and most kids are relaxing after final exams.

She looks very happy playing basketball with other kids, I think this is exactly what I want.

She looked very happy playing basketball with other kids, I think this is exactly what I want.

I see the class as a exercising and entertainment class but not a skill training class, lol.

I see the class as an exercising and entertainment class but not a skill training class, lol.

The Basketball Training Class

It’s correct, but “The” feels a bit weird to me here.

She is training on the basketball court, and I am sitting beside it.

She looks very happy playing basketball with the other kids, and I think this is exactly what I want.

I see the class as an exercisinge and entertainment class, but not a skill training class, lol.

pattern: A, but not B.

I've brought my daughter to her basketball class at a college near our house.

There are only five kids in the class today, because the winter holiday just began and most kids are relaxing after final exams., so there are only five kids in the class today,

She looks very happy playing basketball with other kids,. I think this is exactly what I want.

I see the class as an exercising and entertainment class but not a skill training class, lol.