Now, I'm waiting to watch the live broadcast of the Paris Olympics Table Tennis men's team final, which will happen in 40 minutes--T (that's 9 PM Beijing time).

When adding an aside into a sentence, dashes work as an interruption to insert an interjection or pause, while parentheses are more used to add information to your point. Also, the aside does not need a capital letter at the start.

Luckily, Iit's Friday night, athe perfect time to watch a live broadcast.

"it's" does not need a capital letter, as it does not start the sentence.
"Friday night is the perfect time" is stronger than "... a perfect time." After all, "a" implies one of many, and how many perfect times can there be?
Great work getting the proper form/spelling of "it's." Many native English speakers get this wrong.

The best way to do it is with my friends, but I've just recovered from my latest COVID, and still have a bit of a cough, so I'll watch the game alone this time.

"...from my latest COVID" is a bit awkward; best just say "from COVID." And since the cough is directly related to COVID, you can include that detail within the same clause without using a comma.

The best way to do it is with my friends, but I've just recovered from my latest COVIDCOVID recently, and I still have a bit of a cough, so I'll watch the game alone this time.

"My latest covid" doesn't sound natural. There's generally no reason to use possessive adjectives when talking about illnesses.
The lack of a subject pronouns in "and still have a bit of a cough" doesn't really sound natural either. It can work in some contexts, but I would generally include a subject pronoun. Especially in American English.

Luckily, Iit's Friday night, a perfect time to watch a live broadcast.

The best way to do it is with my friends, but I've just recovered from my latest COVID, and still have a bit of a cough, so I'll watch the game alone this time.

It feels a bit awkward to say “my latest COVID.” I might say “I’ve just recovered from my latest COVID infection” or “I’ve just recovered from COVID for the (first/second/third/etc.) time.”

Now, I'mRight now, I'm (eagerly) waiting to watch the live broadcast of the Paris Olympics Table Tennis men's team final, which will happen in 40 minutes--. That's 9 PM, in Beijing time.

That is to say, I need to stay up late to watch the game.

Luckily, Iit's Friday night, a perfect time to watch a live broadcast.

The best way to do it is with my friends, but I've just recovered from my latest case with COVID, and still have a bit of a cough, so. So, I'll watch the game alone this time.