Yes.

Compare "Danish" as an American term for a Viennese pastry, "he's Danish" doesn't sound odd.

We don't generally abbreviate foods named after places, but:

"Cheddar is 200 miles from both Sandwich and Stilton."

These are all places in England.

I don't see why not. Hamburger means "of Hamburg". If you're thinking about the food item, it isn't a funny coincidence or anything, its name is derived from Hamburg.

I'm not German nor a linguist, YMMV.

Yes. Hence also the names Frankfurter and Wiener (Wien is Vienna, in German).