Some of their words touched me so hard.

SUGGESTION: Some of their words made me deeply emotional.

Quan said that there was something beyond the gold medal, and to her, was to perform ther perfect self.

How great these young kids are!

They focus more on themselves, not than on the outside.

So people cared mucha lot about others' opinions.

Some of their words touched me so harddeeply.

"Touched me so hard" is an unnatural phrase and not commonly used. "Deeply" instead of "so hard" maintains the sentimental nuance of the phrase.

If you don't want to use "deeply", you can write the sentence like this: "Some of their words touched my heart." This is more commonly used
But your sentence would be more natural worded in the passive tense: "I was touched by some of their words." This is commonly used as well

Quan said that there was something beyond the gold medal, andwhich to her, was to perform as the perfect self.

Exchanged "and" to "which" because the second clause explains the first. [To perform as the perfect self] is the "something" which is beyond the gold medal.

"To perform the perfect self" has the nuance that Quan is acting or playing a role (of a perfect self). It has a general or vague nuance. "To perform as the perfect self" has the nuance that Quan is striving to be her perfect self when she performs the dives; it's a more personal nuance.

Chen said that she was lucky to have such a powerful competitor who made her strive to reach a level she had never thought of before.

"Strive to" adds an extra nuance that Quan inspired her to reach the level. It's a more personal nuance again. "Made her reach a level" can imply that Chen physically made her, or told her to reach the higher level.

Another way you can say it is: "Chen said that she was lucky to have such a powerful competitor who inspired her to reach a level she had never thought of before."

How great are these young kids!

To have a complete sentence you need a verb (to be/are).

They focus more on themselves, not on othe outsiders.

"The outside" is non-specific/vague. In order to convey your intended message you would need to add something after outside, like "outside opinions", "outside noise". (If you used these, you wouldn't need "the"). Otherwise, using "others" is fine.

When I was a kid, my parents always talked about the strengths of other kids and my weaknesses, which made me often doubt myself even though I hadwas always been a straight-A student.

The whole sentence is in the past simple tense, so switching to past perfect tense is unnatural.

This kind of method seemedwas very popular in China at that time.

"Seemed" implies that you were observing it happen, but the last sentence said that you experienced it yourself, so it would be more natural to use assertive language.