One of my labmates at the university came from the USA.

Alternative

However, iIt took her a while for her to becaome better at communicating in Japanese.

US alternative

She seemed to havehas a part-time job at an English-language school.

Alternative. In Japanese, we use "〜そう" when speaking about another person's activities. In English, however, we don't use the "seems like" or "appears to" equivalent. This is why it's so tricky for us to learn the other's language.

If I wasSince I'm interested in learning English as much as I am, I would like to ask her to give me feedbacks on my writing or pronunciation.

US alternative

or

Since I'm interested in learning English, I'll ask her to give me feedback on my writing or pronunciation.
Since I'm interested in learning English, I'm thinking about asking her to give me feedback on my writing or pronunciation.

Last week, I came across thean ad ofor the English-language school at which she had workeds.

I can't tell if she still works at the English-language school or not. If she doesn't, then it would be, "Last week, I came across an ad for the English-language school at which she worked" or "Last week, I came across an ad for the English-language school at which she had worked."

I booked thea trial lesson soon after clicking it since it was affordable and reliable.

This sounds more natural in the US.

One of my labmates at the university came from the USA.

However, iIt took a while for her to became better at communicating in Japanese.

For simplicity's sake: However = but = to the contrary. Whatever follows 'However' is typically surprising because you would not guess that it was true after reading the previous information. Here, it is not surprising that it took awhile for her to communicate in Japanese because she could not speak any Japanese when she visited.

She seemed to have a part-time job at an English school.

Seemed to have a job = it appeared that she had a job = you guessed that she had a job. Is this what you mean to say (If yes, this is correct!)? Or, did you mean to say that 'She had a part-time job'?

IfBecause I was so interested in learning English as much as I am, I would ask her to give me feedbacks on my writing or pronunciation.

Usually, when 'if' is used in this kind of context, it expresses something that has not taken place yet (something in the future) and something that may or may not take place. Here, we know that you asked for feedback, so Because or Since make more sense.
"as much as I am" -> "as much as I was", if you want to use this phrase. However, 'was so interested' sounds more familiar.
"feedback" with this meaning (in American English!) is almost always singular.

Last week, I came across thean ad ofor the English school at which she had worked.

"the ad" implies that there was only one ad that existed for the school.
"ad of the school" makes sense, but "for the school" sounds more native.