I haven't slept much this week because ofI have been coughing at night.

This makes it more clear who is coughing at night (you). This could also be rewritten as "because of MY coughing at night", but I think "because I have been coughing at night" sounds more natural.

You know, I've had insomnia since I was a student, but it's mostly gone now, a. Although - I'm still a light sleeper.

Sentence is correct, just quick corrections in punctuation

There's no comparison - being healthy and being able to fall asleep immediately areis so cgoold.

I haven't slept much this week because ofI'm always coughing atin the nights.

However, pneumonia has ruined my sleep.

I found it funny that I got sick after saying that anything can happen to the human body; we're in danger all the time.

Not really a grammar correction so much as a word usage correction, but "we're at risk all the time" or "we're always at risk"

Perhaps what happened to me is just a punishment for my words, but I think this only confirms my statement.

When you are healthy, you don't notice it, you just live.

Perhaps more natural: When you're healthy you don't notice, you just operate/go about your daily life.

It doesn't matter how well or badhow poorly you live, but nothing disturbs you, nothing distracts you from living your life.

So, you can be happy if you are healthy.

Grammar is fine, but consider: So, you're generally happier when you're healthy.
Or if you want the word "if" in there: So, you're more likely to be happy if you're healthy.

There's no comparison - being healthy and being able to fall asleep immediately are both so cool.

I haven't slept much this week because ofI've been coughing at night.

This sentence's meaning is a bit unclear. The way it's written sounds like you're hearing coughing from someone else at night, but my guess is that you're the one who's coughing.

It doesn't matter how well or badly you live, but. If nothing disturbs you, nothing distracts you from living your life.

"but" feels like the wrong conjunction here. Usually, if we say "it doesn't matter...", the next sentence is the point we're making, so this correction makes a little more sense.

There's no comparison - being healthy and being able to fall asleep immediately are so coolthe best.

"cool" isn't the right adjective to describe these actions, it sounds awkward. "the best" isn't necessarily the only description you can use here, but it sounds less awkward.