Does the bun have milk or milk products in it ie: butter, whey, milk, cream, buttermilk, cheese?

It's a normal bun made with white bread, so most likely there is milk in it.

Bun is dairy free, they also have Glutën free buns, or "lettuce Wraps with no buns

No they do not contain dairy. They are made by Franz and the ingredients are available on their website. Source: I work here.

Yes

Yes a normal bun

Yes, they have buns that are buttered.