Typically, the only dairy I find at a Chinese Restaurant is in the Crab Rangoon - cream cheese, I think.
On the Italian front - that can be a good bit more challenging. It's not just pizza without cheese. We've had a situation where we asked that the pizza be made without cheese. It was very lucky that the cook recognized my son (at 18, my sone *loves* to eat out) and told him that there was cheese (Romano, I think) in the sauce.