Matt, 龙龙(Lónglong, his Chinese name), is/was an American young man from southern America that I met just this afternoon.

Because we are/were learning each other's mother tongue, we had a language exchange right after we met for only a few minutes.

When I opened the video of his music box, I heard a beautiful piece of music.

I think ithe recording was played by a piano.

Next, he talked about a couple of Chinese people who were the owners of a shop near Matt's neighbourhood.

He said the couple were nice to him, giving him Chinese tea or some Chinese traditional Chinese snacks.

Matt was very grateful to the couple and he said he'd like to bring some home-made foodfood made by his mother to the couple.

Matt, 龙龙(Lónglong, his Chinese name), is/was an a young American young man from the southern Americapart of the US that I met just this afternoon.

I changed “America” to “US” to be less redundant

Because we are/were learning each other's mother tongue, we had a language exchange right after we met for only a few minutesjust a few minutes after we met.

When I openedsaw the video of his music box, I heard a beautiful piece of music.

This is a bit confusing. What video? You never mentioned a video before.

I think it was played by aon the piano.

Saying it was played “by” the piano makes it sound like the piano itself made the music and not the person who actually pressed the keys.

Next, he talked about a couple of Chinese who were the owners ofpeople who owned a shop near Matt'his neighborhood.

He said the couple were nice to him, giving him Chinese tea orand some Chinese traditional snacks.

I prefer “and” here. It doesn’t necessarily imply that he got both together, just that at some point, he got both.

Matt was very grateful to the couple, and he said he'’d like to bring some home-made food by his mother to the couplethem some of his mother’s homemade food.

He introduced the situationtalked about times in the countryside wheren people helped each other and shared their food with each other, which was also the situation in the Chinese countryside in China when I was young.

He could speak Chinese and his Chinese pronunciation was very good, unot like other westerners who have a strong accents.

When I said February 22 in Chinese, he knew immediately knew what I was saying.

This is not really an error; I just think that “immediately” feels better before the verb rather than after it.

He asked me to pronounce his Chinese name 龙(, which means dragon), in four tones, and he pronounced it right after me.

It was a really niceMy conversation with Matt was really nice.

This word order feels more natural to me.

Matt （(龙龙 Lónglong）

It’s better to use a single-byte parenthesis mark.

Matt, 龙龙 (Lónglong, his Chinese name), is/was an American a young man from the southern part of America that I met just this afternoon.

Insert a space before the left parenthesis mark.

“Southern America” sounds like South America (南美).

Because we are/were learning each other's mother tongue, we had a language exchange right after we met for(session) only a few minutes. after we met.

All of them hadwere very delicate appearances.ly made.

I’m not sure what a “delicate” appearance is. “Delicately made” is idiomatic.

When I openplayed the video of his music box, I heard a beautiful piece of music.

I think it was played by a piano.

The grammar and sentence structure are fine but the logic seems odd. Music boxes don’t have pianos inside them (they’re too small to hold a piano!). Do you mean when the music box was opened, it played a recording of a piano? Usually music boxes contain no electronics, so they don’t play electronic recordings.

Next, he talked about a couple of Chinese (people) who were the owners of a shop near Matt's (his) neighborhood.

He said the couple wereas nice to him, giving him Chinese tea orand some Chinese traditional Chinese snacks.

Matt was very grateful to the couple and he said he'd like to bring some home-made foodfood made by his mother to the couple.

or: bring the couple some of his mother’s homemade food / home cooking.

He introduced thetold me about / described situations in the countryside where people helped each other and shared their food with each other, which was also the situation in the countryside in China when I was young.

In my opinion, we don’t normally use “introduce” in that way.

He could speak Chinese and his Chinese pronunciation was very good, not like other wWesterners who have a strong accent. / have strong accents.

When I said “February 22” in Chinese, he knew immediately what I was saying.

He asked me to pronounce his Chinese name 龙( (which means “dragon”) in four tones and he pronounced it right after me.

Insert a space before the left parenthesis mark.

What do you mean “pronounce his name in four tones”? Do you mean you pronounced it in four different ways, using the four different tones? This seems like an odd request:
long1 long1
lóng lóng
long3 long3
long4 long4
(Sorry, I don’t know how to type pinyin tones on my phone, but I was able to copy what you’d typed for 二声.)

Matt, or 龙龙(Lónglong, his Chinese name), is/was an a young American young man from southern America that I met just this afternoon.

In the order of adjectives in English, age goes before origin

Because we are/were learning each other's mother tongue, we had a language exchange right after we met for only a few minutes / we had a language exchange only a few minutes after we met.

When I opened the video he sent me of his music box, I heard a beautiful piece of music.

Just for more clarity

Next, he talked about a couple of ChinesChinese couple who were the owners of a shop near Matt's neighborhood.

"Couple of Chinese" makes it sound like two random Chinese people who aren't related to each other

He said the couple wereas nice to him, giving him Chinese tea orand some Chinese traditional snacks.

"Couple" is actually a singular noun here, so use "was."

Matt was very grateful to the couplem and (he) said he'd like to bring some of his mother's home-made food by his mother to the couple.

"Them" to remove repetition.
Your sentence isn't wrong but "mother's home-made food" is more natural

Matt said he was born and raised in the countryside before he went to the city. / Matt said that before he went to the city, he was born and raised in the countryside.

Matt, 龙龙(Lónglong, his Chinese name), is/was an American young man from southern America that I met just this afternoon.

Because we are/were learning each other's mother tongue, we had a language exchange right after we met for only a few minutes.

I think it was played byon a piano.

Next, he talked about a couple of ChinesChinese couple who were the owners of a shop near Matt'his neighborhood.

Matt was very grateful to the couple and he said he'd like to bring some of his mother's home-made food by his mother to the couple.

He introducedtalked about the situations in the countryside where people helped each other and shared their food with each other, which was also the situation in the countryside in China when I was young.