What does "affair" mean here? Doesn't it mean "liaison"?

Is there any words that can be used instead of "particular"?

Here particular is being used as an adjective referring to a specific thing rather than others. Therefore some good synonyms in this particular context would be "special", "specific" or "certain".

What's the difference between "refugee" and "exodus"?

A “refugee” is any person fleeing their home (or country) due to reasons of war, religion, politics, the economy, or any other forms of persecution. An “exodus” is not a person but a mass departure or migration, which can include people. Thus, refugees may be part of an exodus, but an exodus is not always caused by refugees. Ex: The number of refugees displaced by environmental change is growing. Ex: Every year, there is a mass exodus of retirees to sunny vacation spots in the South.

How will it be different if I use the word "condition" instead of "demand" here?

If you used "condition," it would be acceptable within this context! However, there is a slight difference in nuance. A "condition" implies a compromise that benefits both parties, even if there are "conditions". A "demand" is the thing that this person really wants. For him, it's the demand or nothing. It's not a compromise. Ex: I'll help you with one condition. You have to buy us ice cream afterwards. Ex: She's demanding to see you. Otherwise, she won't leave.

Why does he suddenly say "breathe it in"? What does "it" refer here?

Maui has made up a narcissistic song about himself. When he says, "yes, it's really me, Maui, breathe it in", "breathe it in" is meant in a similar way to "let it sink in.", which means to 'understand or accept'. "It" doesn't really refer to anything here, the main purpose of this phrase is to exaggerate Maui's greatness and make Moana feel amazement at being in his presence. Unlike in this video, when "breathe it in" is used in the conventional sense, it is used to tell someone to breathe in the air or smells around them. Ex: I love the smell of freshly baked cookies. Breathe it in! Ex: The air is so fresh by the ocean. Breathe in that air!

Is "quite" different from "very"? If I say "very considerable", does the meaning get changed?

"Quite" is similar to "fairly," "rather," "somewhat," and "relatively." This means that the degree is significant enough to take into consideration, but not to the point of "very," which is stronger. Ex. This shirt is quite nice. (This shirt is rather nice.) Ex: Work has been quite smooth this week.