Because one of my sons is infected by the COVID-19, all of my family members shouldmy whole family has to stay at home for a week.

"all of my family members" is okay, I just feel like "my whole family" is a more natural way to say it.
"should" is too lax, it feels like a suggestion.

My wife and I did the remote workworked from home, and my first son, who is in an elementary school, studied remotely.

This was the first momenttime that my son studied remotely atfrom home.

So we weren't familiar with the system, and we faced many issues during classes.

Fortunately, we finally finished all classes of this week's lessons without any big accidentissue.

Tomorrow, we will check again, whether infected or nottest for COVID again.

I hope that no one has a problem on the testtests positive.

Because one of my sons is infected by thehas COVID-19, all of my family members shouldhave to stay at home for a week.

My wife and I did the remote workworked from home and my first son, who is in an elementary school, studied remotely.

This was the first momenttime that my son studied remotely (at home).

Fortunately, we finally finished all classes of this week without any big accidentproblems/issues.

Tomorrow, we will check again, whether my son is infected or not.

or if you are all ill "whether we are infected or not"

I hope that no one has a problem on the testtests positive.

or "I hope no one has the virus

Because one of my sons is infected by withe COVID-19, all of my family members shouldmust stay at home for a week.

It isn't wrong to say, but ‘must’ or ‘have to’ sounds more natural👍

My wife and I did the remote work, and my first son, who is in an elementary school, studied remotely.

It might sound more natural to say, 'we worked remotely' OR 'worked from home'.

You can also say, 'switched to online learning'.
We'll usually say 'is doing online learning', 'took online classes', 'took his classes online (from home)'.

This was the first momenttime that my son has studied remotely at/from home.

You can also say, 'This is the first time that my son has tried online learning (from home)'.

Fortunately, we finally finished all classes of this week without any big accidentmishaps.

To express this idea, we can say 'without any mishaps' (formal) OR 'without a hitch' (casual)👍

Tomorrow, we will check again, whetherto see whether we are infected or not.

I hope that no one has a problem on'fails' the test.

You can also say:
- I hope we 'pass' the test.
OR
- I hope our test results are negative.