What does “check someone out” mean? Is this phrase only used in flirting situations?

what's 'buried' mean here?

Good question. It may seem confusing to use "bury" in this context. However, one of the definitions of "bury" is "immerse," "absorb," or "engross." You might hear sentences like: Ex: She buried herself in her work. Ex: He was buried in housework throughout the week. We often see "bury" used in combination with "head" or "face," especially in the context of reading. In this case, the speaker is saying that he was engrossed in/concentrating very hard on reading/looking at the map, so his chin was "buried" in it. Ex: His head was buried in a good book. Ex: That book must be very interesting, considering how his face is buried in it.

What does "doing" mean here?

"Doing" in this case means to "pick" or "choose". This is not a very common use of the verb "do". The word "doing" instead of "picking" or "choosing" is most likely used in this case to sound different and unique. Here's another example of "do" with the meaning of "choose": Ex: I wanted to do that topic for the project but she's already doing it.

Is it okay to say "outside something" without "of" like "outside of something"?

Yes! It is perfectly acceptable to omit "of" in sentences like these and still be grammatically correct. However, this can't be done in every situation. It's a case-by-case situation. Here are sentences where you can remove "of." Ex: I'm outside the gate. Ex: He lives just outside of New York City. Here are sentences where you can't remove "of." Ex: We are outside of the Appalachian mountains. Ex: She is outside of the lake.

How come "like to tell" was used not just "told" here? Is there any difference between those two?

"Liked to tell" in this case implies that someone often told, or enjoyed telling people about something. This is slightly different from just saying "told", as this doesn't imply frequency or emotion from the action of having "told" someone something. Since "liked" is the main verb in the phrase "liked to tell", "liked" is use in the past tense here, and "told" becomes "tell".

Does "awesome" have same nuance as "good"?

Yes, the nuance of "awesome" is similar to "good", but too an even more extreme extent. "Awesome" implies that something is better than "good" and the nuance is more similar to "amazing". "Good" just means that something is not bad, and sometimes the nuance of "good" means it is just okay or so-so. Ex: This book is awesome. Ex: This book is good. In the examples above, the first sentence implies that the book is better than than the second sentence, as "awesome" makes it sound like the book is really great or amazing. In the second sentence, we get the feeling from the word "good" like the book is just okay.