Question: It is a frequent idiom for a group of characters to interact while eating Chinese food. Generally of the carry out variety, with chop sticks being plunged into the deep fold-up boxes. It seems to me that this is intended to bring a particular thematic element, but I can't quite put my finger on it. I see it way too often for it not to have a specific significance.

Answer: 
I do think you have something in thinking that Chinese food has a particular significance as take-out. There's an aura of a comfortable, but pressured, environment, where people are banding together to get stuff done. But, enough of my personal opinion and on to some quotes and particular examples. Much of this is cribbed from TV Tropes. 

* **Jewish people and Chinese food:** Starting with the trope of Jewish people going out for Chinese on Christmas ([Peking Duck Christmas](http://tvtropes.org/pmwiki/pmwiki.php/Main/PekingDuckChristmas)), there is a broader association in all cases. For example, in *My Favorite Year*, the more urbane of two put-upon assistants bonds with the romantic interest by ordering Chinese food when they have to work through the night, saying "Catherine, Jews know two things: suffering, and where to find great Chinese food."
* **Chinese food and long hours/stuff to do:** This is particularly associated with *The West Wing* where the characters order Chinese food when they're facing a long creative challenge, like writing jokes for the Correspondent's Dinner
* **Chinese food and comfort:** *Orange is the New Black* uses the theme this way by centering the main character's romantic choices around a quote about Chinese food. After having a wild and tempestuous relationship with Alex, a drug smuggler, the main character Piper is told by her more settled best friend that eventually, she's going to want "someone who knows when to order Chinese food".

In summary, Chinese food is associated with familiar but stressed situations, particularly among more intellectual and/or Jewish characters; and can imply both "something needs to be done" and "something needs to get fixed emotionally".
I think that movies and TV shows use those Chinese takeout boxes just because it’s easy for production. They could be empty boxes and they wouldn't actually have to prepare any food for the scene or multiple scenes at that.