For someone who doesn't know what Teockguk is, it's a Korean traditional dish toKorean dish that is eaten on / that we eat on the first day of the year.

You could also just say "eaten on"
In my opinion, the best option is always the active voice "that we eat on"

We believe that if we eat a bowl of Teock-guk, we will get one year older.

So when I was younger, I ate many bowls, because I wanted to grow up faster.

When "because" is in the middle of a sentence and what follows is essential, no comma is needed.

I wanted to focus on drawing today, but felt super lazy!

"but felt super lazy" is not a complete sentence or an independent clause because it doesn't have a subject. For this reason no comma is needed.
A comma is needed here: "I wanted to focus on drawing today, but I felt super lazy!"

Maybe it would help to think of the comma as a cut you make with scissors. After you cut the sentence into two parts, if both pieces have a subject and a verb, then a comma goes between them.

I gave up being productive and decided to just take it easy.

Here you punctuated the sentence correctly. It's the same pattern as the sentence before it.

I'll be back studying early tomorrow early morning!

You could say "I'll be back studying tomorrow morning early," but that doesn't sound as natural to me.
Another option is the idiom "at the crack of dawn"

I stayed up late to greet everyone last night, so I slept in this morning.

Grammatically fine, but without context it's not clear what "greet" means here. Without context, I would assume you had guests to greet, since that's the nuance of greeting.

For someonthose/people who doesn't know what Teockguk is, it's a Korean traditional dish to[commonly] eaten on the first day of the year.