My cousin got an internship and was ready to take it.,

But because he was alone and, he left early to get to Wuhan from another province.

The sentence I corrected has a slightly different meaning, sorry! I wasn't 100% I understood.

He tells us how his quarantine goes onis going from time to time and h. He is good and healthy.

Somehow oOne day, just as the officials declared that in two weeks, Wuhan city would no longer be in lock down two weeks later, which, meansing that my cousin will be no longer be under quarantine and he can go back home, my aunt was inhad an argument with my father.

My father suggested that my cousin remain in Wuhan since Wuhan has all the attention from all over China due to COVID 19, there will be great development potential with favorable policies.

She just wanted her son to come back as soon as possible.

ButI don't understand how she could she evern think about it?that.

My correction means the same thing but sounds a little more natural. :)

You cold also say "How could she even think that?" if you're talking about your thoughts.

Our family was bornis from there.

And sShe spent 18 years there.

Nobody knows why she said what she said.

He willdecided to stay in Wuhan.

ButHowever, he told us theat his company closed down and would not longer offer anyone work.

Since it's his decision, he had tos been struggling hiswith life and must find other ways to live on.

But he was alone and early to get to Wuhanit was in Wuhan, and he was alone and arrived early from another province.

"Early to get to Wuhan" doesn't make sense here, since you haven't introduced the role of Wuhan yet in the writing. It seems like something is missing

So he has been stuck in the epicenter of the coronavirus outbreak since the middle of January.

He tells us how his quarantine goes on from time to time and h. He is good and healthy.

"And" must connect related phrases. You could also write "and says that" or something similar, but if not, this sounds disjointed. "Goes on" is related to continuity (e.g. "the adventure goes on"). To say how something is, just use "go."

Somehow one day, just as the officials declared that Wuhan city would be no longer be under lock down in two weeks later, which meanst that my cousin will beould no longer be under quarantine and he canwould be able to go back home, my aunt was in an argument with my father.

"Two weeks later" sounds like you are referring to when the official made the announcement in reference to the rest of the story, not when lockdown will be lifted.

My father suggested my cousin remain in Wuhan: since Wuhan has all the attention from all over China due to COVID 19, there will be great development potential with favorable policies.

You've introduced a subordinate clause with "since" that does not fit anywhere and creates a run-on. You need to either separate the sentences or reword it, like "My father suggested my cousin remain in Wuhan because there will be great development potential … since Wuhan has all the attention."

She considers that city a hell.

But how could she ever think about ithat?

"Think about something" means that you're thinking about a particular subject. To say that you're thinking a particular idea, "think that" ("that" being a pronoun referring to the idea) or "think that…" (to introduce the idea itself) should be used.

Our family was bornis from there.

People and individual things are "born," not families.

But he told us the company closed down and would not longer offer anyone work.

Since it's his decision, he had tos been struggling with his life and has to find other ways to live on.