I wouldn’t have to. Don’t know about OP. He mainly talked about not wanting to eat meat. Dairy can be present in very small quantities in many dishes, as you know. But it’s not like I go eat eggs and drink cartons of milk. Anyway, even this is rarely a problem in my experience. If I do eat such things it is out of politeness to the donor who is usually aware of me being vegan but unaware of what veganism means.
I’m not wandering in Thailand nor does the OP seem intent on doing so, so this is somewhat irrelevant. Regardless, monks having done so and have refused meat. Since there is no Vinaya rule against us doing so. For OP, I explained a bit further here.
Not in many places I’ve been. But it’s often possible to be vegan anyway, if you’re not too picky or need many different choices.