"Cake" in Korea only means one thing, birthday cake. But does it have broader meaning than that?

Why "way" is used in here? I think it is okay even though if it is not in here.

Yes, you're right. 'Way' in this case is used as an adverb meaning "a lot". Ex: I was way ahead of him.

What's "back in the days" mean?

"Back in the days" is an expression used to refer to a time in the past, usually a period in history. In the case of this video, it is describing a time before people commonly slept on mattresses. It's most often used with a descriptor following it that explains the time period. Ex: Back in the days of no electricity, people used candles for light. Ex: Back in the days of feudalism, people often had short lifespans. A similar expression, "back in the day" (without the plural "days") is used to refer to a time in the past, but is more often used in situations in which one is recalling or looking back on the past fondly (usually used when talking about one's childhood or youth). Ex: Back in the day, we used to ride around the city on bikes. Ex: Back in the day, I used to go to the pool with my family in the summer.

What's the main difference between "revenge," "retaliation," and a "vengeance?"

Good question! The phrase "with a vengeance" means the degree to which something is true, "powerfully" or "forcefully." "Revenge" is the action done, "vengeance" is the noun to describe the action of revenge. "Retaliation" is the return of an attack, it doesn't necessarily have the same intention as "revenge" to inflict pain on something done wrong, it's just fighting back. Although, "retaliation" can also be a form of "revenge." Ex: When someone yells at you, don't retaliate by shouting back. Ex: The new rule is a retaliation from the teachers since a few students got in trouble yesterday. Ex: I'll take revenge by pranking him. Ex: I want vengeance for the pain he caused. Ex: She reentered the competition with a vengeance.

I haven’t heard of “unto” before. Is it still used?

No, it is not still used, "unto" is Middle English. It can be found in old English texts like the Bible, or Shakespeare!

Does "individuals" sounds more formal than "people?" If so, would it be more appropriate to say "individuals" on formal occasions?

It's similar, yes, but! "Individuals" emphasizes singular people, rather than as a whole. So, saying "people" may just mean a group or many people in general, but "individual" emphasizes each person in particular. A good synonym here would also be "each person," but you're right, "individual" does sound a little more formal! Ex: I got everyone individual presents. => One each Ex: Each person had something to say. => Here, you couldn't use "individual." Ex: I supervise individuals in the company.