In an out-schoolextracurricular / a supplementary class with my daughter

"an out-school class" is not a standard type of class. I have never heard that term in English.

My daughter and I go to an out-schoolextracurricular /a supplementary class to study mathematics every Sunday morning.

"...go to an extracurricular math class..." is another way to say this in fewer words, and more informally.

We get up at 7AM a.m. and arrive at the class at about 8AM, we finish the classround 8 a.m. The class finishes / is over at 10:20AM a.m.

"about 8 a.m." and "around 8 a.m." have the same meaning, so you choose based on how it sounds. I think "around" flows better.
There are different conventions for the time, and I chose what the internet told me is most common. We often use "is over" for a class.

My daughter is ain grade sixth primary school student.¶
My daughter is a sixth grader.

The other words are not needed.

Most content of the out-schooextracurricular / supplemental course is harder than what she studies in school.

I really have no idea whether studying in out-schooltaking this class is useful for her future.

This is the way a native speaker would probably word it, and it is succinct.

But we have to take the class just because of the involutionary? education atmosphere.

"involutionary" is an unusual word with a narrow meaning. I had to look up the definition, as it was a new word for me. I suggest using a more common word that a general reader would understand. Even after reading the definition, I'm not sure what you mean. Based what I know about the Chinese education system the first word that comes to my mind is "competitive."

Nearly eachvery primary school student takes the out-schooextracurricular / supplemental classes.

"Nearly every" is correct.
"Almost every" is also correct.
("Nearly each" and "Almost each" sound wrong.)

"the class" means they take the same class. This seems unlikely. If they take a variety of classes, then the plural or an/a is correct.

We get up at 7AM and arrive to the class at about 8AM, we finish the class at 10:20AMthe class finishes.

ok so very small error with the "to" but "we finish the class" sounds like it is something you have the ability to change the length of. in english "finishing" something in the simple present of "finish" like "i finish" or present progressive like in i'm finishing" sounds like something you have control over you finish your homework but your class " finishes" by itself you don't control when it finishes the class " chooses" when it finishes

My daughter is a grade sixthsixth grade primary school student.

just how we say it in the US but it could be different in other places

In an out-schoolextracurricular class with my daughter

My daughter and I go to an out-schoolextracurricular class to study mathematics every Sunday morning.

We get up at 7AM and, arrive at the class at about 8AM, weand finish the class at 10:20AM.

My daughter is a {grade sixth | sixth grade} primary school student.

Most content of the out-schoolextracurricular course is harder than what she studies in school.

Some content is what I studied in {middle school | junior middlehigh school}.

I really have no idea whether studying {in out-school class is useful for this extracurricular class | in extracurricular classes} will help her in ther future.

But we have to take the class just because of the involutionary education atmosphere.competitive nature of our educational system.

As I’ve mentioned in my corrections to some of your other posts, we don’t use the word “involution” in English, except in some obscure scientific contexts.

Nearly eachvery primary school student takes {the out-schoolis extracurricular class | extracurricular classes}.