For your first question I'm going to show you this very relevant post about Roman Snack food. (i.e, fast food.)

I'm more knowledgeable about the medieval Era so I'll answer that question. In urban areas, yes, street food was sold in a manner which you could describe as fast food. In the streets of major cities like Vienna you could find mobile ovens serving meat pies in the streets. and in the book Le Mangier du Paris you could go to the market to buy ready made sauces. Simple wafers were also sold by street hawkers as well.

If you want to learn more about ancient cookery you could visit my new sub, /r/archaiccooking