What does “shine up” mean?

What does the "penthouse" mean? What makes it different from ordinary houses?

A "penthouse" is a luxury apartment on the top floor of a building, usually with a nice view. It's typical for hotels to have them, too. It's different from an ordinary house since it's an apartment and it's very luxurious. The top floor of a building with a nice view is considered very desirable. Ex: One day, I want to have my own penthouse. Ex: I don't get why people like penthouses so much. I'd prefer a little garden cottage, honestly.

Isn't it supposed to be "earlier?" Not "early?"

Not quite! "Early" means before an expected date or time, whereas "earlier" means "more early." Ex: You're earlier than I thought you would be. = You arrived earlier than the already early time I expected you'd be here. Ex: We'll go a little earlier just to make sure we're there on time. Ex: I don't mind arriving early to parties.

What's "coming of age" mean?

"Coming of age" is a stage when a person or thing becomes an adult or fully mature. In most cases, it's used to describe when people become an adult (not necessarily legal age). Ex: His book is a coming of age story during wartimes. Ex: I have fond memories of when I came of age.

Can I assume "look out" and "watch out" as a same meaning?

Yes, you are right, the expressions "look out" and "watch out" both have the same meaning of warning someone to be aware of something, and they can be used interchangeably!

Why does he use 's and say "friend of Rachel’s", not "friend of Rachel"? I know that I should put 's when I'm saying "Rachel's friend".

He says "friend of Rachel's" because something used to follow after the person's name (here, "Rachel") that is now omitted. In modern English, it is not known what used to follow after someone's name in this phrase, so it does not make sense to use " 's" here. Therefore, "friend of Rachel", or better, "Rachel's friend", should be used. However, both "friend of Rachel's" and "friend of Rachel" are considered grammatically correct.