I have a pal in Florida, an older American, who helps me a lot with my English.

Old is grammatically correct, but old people will get mad at you if you call them old. It's typical to say "older" to be a bit more... diplomatic... ;)

David didwasn't learn anythinging a language himself, he simply spent his time helping learners from all over the world with studying English.

He "didn't" learn anything is more about the fact that he did not learn a single fact/new thing/etc than that he just wasn't on iTalki to practice a language himself. The rephrasing makes it clear what was going on and is a bit more stylistically elegant (at least to my ears).

David was quite talkative, as anyre most retired personople, and I often listened to him, while thinking deep down "Oh, man, don't you want to pause for a second to give me a chance to say a word or two?"

the edit to retired people is grammatical, the other is stylistic.

Later on, I realized that I should have simply stopped him if I wanted to have that chance, and things got better.

Just stylistic changes to make things sound more "typical."

When I was in the States - —the trip, which lasted for 8 months - —David often helped me with his suggestions, and a. A couple of times his tips were really useful, they worked out.

The dash you're looking for here is the "em-dash" —. It's the longest one, and there is no space after or before it.
"They worked out" is a bit redundant. Splitting into two sentences is better style.

ByAt the end of my residencevisit/exchange/program/time in the US, he invited me to his house in Florida, but, unfortunately, I wasn't able to visit him.

I gave a few options for replacements for "residence," which isn't the right style/usage here.
At is better than by for the beginning.

Messaging became my favorite way to practice English, since I was allowed tocould message him 24 hours per day.

not sure what "practice" means. If it's not specifically about English and more about a fun way to spend time, "pastime" works well.

I learned a lot from his messages, and the things that I learned were real American language - —with slang, idioms and expressions you'd never learn from textbooks.

the em-dash again!

At the moment David doeisn't responding to me, as a month ago he took a decision of havingdecided to go on a lengthy meditation retreat - —the longest he'd ever had.

David didn't learnstudy anything, he simply spent his time helping learners from all over the world with studying English.

"He didn't learn anything" could mean he was close-minded.
But as I just broke the parallelism that you had ("learn" ... "learners"), let me suggest a way to recover i: you could say "helping language students from ..."

We had numerous talks on Skype during which my friend told me about his family, American politics, movies, et ceterac.

What you wrote is not wrong, but I think it is uncommon to spell out "et cetera"

Later on, I realized that I should have simply stopped him to have that chance, and things got better when I started doing so.

I think "later on" is a bit too informal (spoken language) and that simply "later" sounds better here.
Also, I speculate that things probably got better when you actually changed your behavior, not just when you realized that should do so.

When I was in the States - the trip, which—on a trip lasteding for 8 months - —David often helped me with his suggestions, and a couple of times his tips were really useful, they and worked out well.

Or "When I was in the States— (for about 8 months), David..."
The "they worked out" is a sentence fragment, and needs some connecting word to the main sentence, or needs a preceding semicolon.

Messaging became my favorite practice, since I was allowedble to message him 24 hours per day.

"Allowed" sounds odd here, as it raised the question as to who was allowing you to do so. If you mean that David allowed it, then I suggest "since he allowed me to message him 24 hours per day" to make that more clear.

I learned a lot from his messages, and things that I learned were real American language - —with slang, idioms and expressions you'd never learn from textbooks.

My correction is terribly unimportant and unnecessary; i merely changed a hyphen to an em dash. An em dash is the appropriate punctuation here, but as regular keyboards do not have it, it is extremely common to use a hyphen in its place.

So I'm only mentioning this unneeded change in case you're interested.

At the moment David doesn't respond to me, as a month ago he took a decision of having a lengthy meditation retreat - —the longest he'ds ever had.

Changed this from past perfect to present perfect because it's still going on, I infer