Marriage is not for everyone, no. Even for many who desire it, it may not actually be for them. Such is life that we often do not know what is best for us, and even when we know what's best, sometimes we still do not take appropriate action—but I digress.

People change as time passes. Is it any surprise that two people who are compatible today might not be compatible months from now? What about years? Decades?

This idyllic pastoral scene is the only part of the picture.

If you used "the" in this sentence, it would be incomplete. "The" would require more details.

"This idyllic pastoral scene is the only part of the picture we will ever see." (Maybe because the farmers in this region are very protective and so do not allow outsiders to visit.)

Without any explanation, it must be an indefinite description, so "the" is not appropriate.

Another way to think about it:

”只是（照片）的一个部分“ => no "the" 。 。 。 "only a part of (something)"

”就是（我们能看得到的照片）的一个部分“ => "the" 。。。 "the only part (that we can see)". 

I will respond over several days as there is a lot to discuss in this question. You may have better luck if you post your questions individually instead of all your questions together.

The

1. Correct.

"The" can modify "most" when it means "最多”, not when it means "多半儿“.

2. Incorrect.

What kind of cost is being discussed? <em>The specific cost</em> of salvaging a sunken ship.

3. Correct.

The seven feet of water is here an indefinite water. For contrast, you would need "the" if they had been more specific about which water: ". . . in the seven feet of water of the Tokamak River."

4. Correct.

 This is the same reasoning as #3.

P.S. "Big guy" is not an important thing to know, so don't worry about this one too much. :)

Short Explanation

"Big guy" is often used to give encouragement or to support somebody who is having trouble. However, it is most often used by adults speaking with children, so when it's used used among adults, it can be patronizing. In this episode, though, it is not meant in a patronizing way.

Example

Son (crying): "Dad, Blake just stole my juice then told me to go away . . . "

Father: "Hey there, big guy! I can get you more juice, but let's go talk with Blake and see what's going on."

Explanation of Script

[My additions are in bold.]

(Joey enters.) I'm guessing Joey just got back from an audition.
All: Hey! Big star! Everyone is trying to be supportive.
Monica: Hey, wait a minute, aren't you the guy who plays the butt in the new Al Pacino movie? Monica is joking, asking if Joey has a role in an important movie (Al Pacino a <em>hugely</em> famous actor).
Joey: Nope. 
Ross: No? What happened, big guy. Ross is trying to be supportive by saying "big guy".
Chandler: (to Ross) "Big guy?" Chandler finds this odd because the term is usually used toward children. 
Ross: It felt like a 'big guy' moment. Ross says that he felt that using the term was appropriate. In fact, I agree with Chandler that it was an awkward thing to say.
Joey: I got fired.
All: Oh!