What does the "country club" mean? What makes it different from ordinary clubs?

Is the subject skipped in this sentence? I can't understand the structure of this sentence.

That's right! The subject is skipped in this sentence because it's implied. This is a common occurrence in English and is usually implied within the context. So, here, they're talking about the bedroom, so Stuart is referring to the bedroom as well in this sentence. If it were a full sentence, "it" would be good to use here since it refers to something already mentioned. Ex: Really is cold today. = It really is cold today. => "it" means "the weather" Ex: Sunny this morning. = It's sunny this morning.

What does the "country club" mean? What makes it different from ordinary clubs?

A "country club" has a lot of amenities, and perhaps a golf course. It's known to be prestigious. An ordinary "club" may not have all the same things and may not require as much of a fee to enter or be a part of, unlike a country club. Ex: I go to my uncle's country club on the weekend for good food. Ex: I'm going to chess club this afternoon! Want to come with me?

If I change this sentence to "I am working for Emirates as a first class crew.", is "a first class crew" less emphasized?

That's a good question! In general, whatever is put at the end of a sentence tends to have less emphasis since it is heard last. So if you changed the sentence order here and put "first class crew" at the end of the sentence, it would slightly reduce the emphasis on "first class crew" and cause "Emirates" to be emphasized a little more strongly. However, this change in emphasis is very minor and the overall meaning of the sentence would still remain the same if you did this.

What's "look behind" mean?

It would actually be more natural to say "looking back," which means "in hindsight," to reflect on past events. Although "looking behind" seems similar in meaning, it's actually incorrect. Ex: Looking back, I made a lot of mistakes when I was younger. Ex: Looking back on your life, what is your most favorite memory?

About the verb "believe", what's the difference to say between "I believe you" and "I believe in you?"

We say "I believe you" when someone tells us something and we are sure it is true. "I believe IN you" is said when we are sure someone can do something (usually something difficult). Ex: I believe you when you say you didn't steal my dress. Ex: You can win this race, Tony. I believe in you.