So what's the deal with small Chinese restaurants? They seem to give you some hot water but no tea. Is it acceptable to bring your own tea? What about other soft drinks? Would the proprietor care if I brought in a Coke Zero, if they don't sell it? They never seem to :-).
You really should not be drinking any soft drinks, its so bad for you whats in those, bad for immune system and your diet, Coke Zero is a can of cancer.
The small little eateries that are everywhere in China? Of course they don't care if you bring in your own soft drinks. If they're really stingy, they might complain if you bring in a soft drink that they sell there, but that's pretty unlikely. I don't think it's ever happened to me before. Restaurants are like everything else in China - there are no rules.

Awesome! Thanks for the feedback. I bought some tissues as there never seem to be any napkins either ;-).

I have never seen a restaurant where bringing your own drinks was a problem (even when they had the same thing for sale) . Even in fancy restaurants, my friends usually bring their own baijiu.
But they will give you tea if you ask for it.
Most small restaurants are not going to care, particularly if you are a foreigner. It would be far easier to bring in a coke zero than to get them to take the peanuts out of their kungpao chicken.
Do understand though that you may be charged for beer or sodas you bring in. Particularly small restaurants, do not keep a detailed bill. They just add up whatever is at your table. So an empty bottle of coke zero might add 3-5 RMB to your tab, even if you brought it in with you.