Ef-s do not work with full frame.
The real difference I see from my L and ef lens, would be build quality. And typically perform better wide open (2.8 , 1.2 ... ) stopped down to say 11, quality becomes similar in real world use.
Your third question is a little harder, ef and L lens are meant for full frame, only because they will let you use the while image area of the sensor, whereas ef-s lens normally have "smaller" glass, because they have less sensor area to cover. Most ef or L lens will actually appear better on a crop camera because the camera only uses the center portion of the lens, which in most cases, is better.
3rd party lens makers have a variety of different ways of labeling the lens for crop or full frame. Best advice, since there are so many different types, is to check online per lens that you are considering. especially the wider lens, like 10mm. Most of those are crop only.