I'm suffering from mild fever coming fromafter the second dose of Moderna vaccine.

"coming from" makes sense here, but sounds quite unnatural.

Our company allows employees to take a special leave, so that I stayed at home today.

Some people could have severe side effects such as headache or dizzyiness, which makes me realize me how potent the vaccine is.

Dizzy is the adjective, dizziness is the noun, the state of being dizzy. When you suffer pain, it is painful - when you suffer dizziness, you are dizzy.

However, we have to keep taking care of ourselves, because the vaccine areis less effective to dagainst the Delta variant.

Or "the vaccines are". Both would be fine, just have to match plural.

Even so, two doses of vaccines are able to neutralize COVID-19 symptoms, if not prevent infection itself.

"vaccine" here is uncountable. You could say something like "two doses of one of the vaccines", but that's quite awkward.

I'm really worried about my father having been coughing all day.

I think this is probably grammatical, and I can imagine a native saying it. But it would be less awkward as "I'm really worried about my father, who's been coughing all day" or "I'm really worried that my father has been coughing all day".

The sudden rise of COVID-19 cases in Tokyo areis depriving residents of opportunities to receive healthcare in a hospital.

"The rise" here is singular, even though it's referring to a plural number of cases.

Even if you were suffer from common diseases other than COVID-19, you'd have to stand the pain and fear ofor getting worse.

"fear for" is more like "fear on behalf of someone or something". "I fear for my son, who is fighting in a war." "I fear for my future, I want it to be good but it might be bad." Fear for [something]: I'm worried that [something] might go badly. Fear of [something]: I'm worried that [something] might happen.

"you'd" is "you would", second conditional.

What all of us hope for must be the end of COVID-19 pandemic.

I can't decide if this is okay or not, but I quite like it. You could say it more simply, "We must all hope for the end of the pandemic", "The end of the pandemic must be what we all hope for". But they sound less interesting than the way you've said it, it's sort of poetic.