one confusing part from "The Terminal"

[Joe]

Amelia Warren: ...(sobing)You know the worst part of it? I never asked him to leave his wife. I was encouraged him to get counseling...I'm rooting for the home team. I just wish the sex wasn't so amazing...You know, sometimes in the mornings, I just stare at him over room service. Watch him do the crossword puzzle. I start to think...may this could happen that we belong together.

Navorski: This man has you. Why he needs...puzzle?

Amelia Warren: This is my problem. I always see men the way I want to see them.

(Warren has a love affair with a married man, with whom she just got off the phone. She is quite unhappy. She's talking to this Navorski guy about her and the man. )

I'm not sure of specific meanings of the bold parts.

home team: referring to the man and his wife???
room service: ???
do the crossword puzzle: I guess it does not mean what "do the puzzle" usually mean. Then, what is it?

Could you please explain to me? Thanks for help.

[Joe]

Haven't I given enough context for you to understand the dialog, or is that something like a "taboo" subject on this forum? Whatever it is, please tell me. I think I could keep that in mind next time when I try to put up a post.

[Marylin]

Originally Posted by Joe

Haven't I given enough context for you to understand the dialog, or is that something like a "taboo" subject on this forum? Whatever it is, please tell me. I think I could keep that in mind next time when I try to put up a post.

Didn't you ask the same question before and got no answer from anyone?

We answer questions regarding English grammar, phrases, idioms, etc. Questions regarding life ethics are not our primary concern here as long as you behave and stick to the rules of the forum.

Having said that, I will explain what I can, even though I have not seen the movie.

I am rooting for the home team.

A home team in sports is a team that is the host of a given sporting event.If it's Seattle Seahawks playing against another NFL team, their home is Seattle. If I live in Seattle, I will be obviously rooting for Seattle because they are my home team too.
Home team here means a family, one unit.
Watch him do the crossword puzzle -
Why he needs...puzzle

stare at him over room service...I am not sure about that one. The room service comes in and she still keeps staring at him unaware of others being there?
He asks her about the crossword puzzle because maybe he feels that she should have his undivided attention. If they meet in a hotel and he keeps himself busy doing crossword puzzles instead, it doesn't sound like he is trying hard enough to make her his priority.
Again, it's a guess. I haven't watched the movie.

[Joe]

Thank you very much. Marylin, I've asked questions like that before, but I've never got unanswered. For example, I used to put up part of a political report, asking what the politician was really saying, and someone else on the forum gave their opinions. I didn't see much of a problem here. I'm learning a language, but not studying linguistics. And we use language in every possible context, including that of life ethics.

Don't get me wrong, Marylin. I appreciate your help and advice. I just get a bit confused.