It would depend on the type of food. If you cooked spices in a frying pan without oil, for example, it would be called 'toasting'. It could also be dry frying: Dry-Frying means cooking food that naturally contains oil or fat in a dry frying pan without adding additional oil.

I think it's always worth looking at a few different dictionary definitions of a word. The Google (Oxford) definition is "cook (food) in hot fat or oil, typically in a shallow pan." but Merriam Webster say "to cook in a pan or on a griddle over heat ESPECIALLY with the use of fat". Personally I would use this second definition. For me frying doesn't always involve fat (oil, butter, etc) BUT describes the method of cooking where something is cooked on a flat surface (usually metal) with heat from underneath. Therefore, I would say "fried tomatoes" in the example you provided. HOWEVER, there are some strange exceptions which are related to the cooking process, but hard to describe. For example, if we put an egg in a shallow pan with heat underneath with fat then that is frying, if we use water instead of oil it is poaching, but if we use those eggs to make an omelette then we don't usually say either 'fry' or 'poach' (we say 'make' or 'cook')! :-)

Yes it is still frying. You might just say "gently fry" or "simmer" on a gentle heat to get the message across that you have no oil or fat and do not want to burn the tomatoes. It is very common to fry tomatoes with no oil just relying on the juices in the tomatoes to prevent burning.

I would just say they are cooked tomatoes. The American dish ‘fried green tomatoes’ is always cooked with oil or fat to my knowledge. In common usage here ‘frying’ implies oil or fat and high temperatures. Even adding a little oil for taste wouldn’t necessarily make it ‘frying’ in my opinion.

fried if you used a frying pan no matter oil or butter or pig's fat