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 don't think Chinese food, in particular, is being used as a theme. I think it's delivery food in general. It evokes the sense that these people don't have time to go anywhere or do anything other than work on the project they're currently working on.

In America, there are really only two universal foods that are delivered: pizza and Chinese. That is changing, and it varies by region, but those are the two standards.

So, the question then would be, "Why choose Chinese over pizza?"

1. Sometimes they don't. Sometimes it's pizza.
2. Some people might considered Chinese a "higher class" of food than a greasy pizza.
3. (My personal opinion) When the characters have that A-HA! moment, it's much more effective visually to point a pair of chopsticks while saying, "Yes! That's exactly it!" than it is to point a floppy piece of pizza.
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.