Do sharks spend a large percentage of the time swimming at the very top of the water, such that their fins can stick out? If so, why? (It doesn’t seem that way from the sharks I’ve seen in aquariums.)
If there’s prey at the surface (e.g. a hairless ape), the shark is likely to be at/near the surface; I think that’s when you’re likely to see fin action. Otherwise, no, I wouldn’t imagine they have any particular reason to be there.
A counterexample would be the great white sharks off the coast of South Africa, where they deliberately attack their prey - seals - from behind and below so as to catch them completely by surprise. No fin showing, then suddenly a thousand pounds of airborne shark.
No, they don’t. This is mainly just a movie trope. However, if they are feeding on schools of fish near the surface, or in very shallow water,the fin may break the surface. See here for more discussion.
If you’re showing a dramatic scene of a shark coming in for the kill, then it helps the movie to show a fin slicing through the water, because otherwise the audience just sees boring, flat, still water. It’s important for movie directors, less important to sharks in real life.
So to answer your question more precisely: people spend a large percentage of their time watching sharks with their fins out of the water. The sharks themselves don’t spend much time that way.
What inspired my question - beyond the “movie trope” - was a report I heard on the radio that a local beach had been closed down because shark fins had been observed sticking out of the water. But after a few hours the beach reopened, and I assumed this was because they no longer saw any fins. (Otherwise how would they know the shark had left the area?)
Well, they do sometimes swim with their fins out. But although seeing a shark fin demonstrates that a shark is in the area (assuming it is correctly identified), not seeing a shark’s fin does not demonstrate that a shark is not in the area.
I believe basking sharks are known for doing this and are rather impressive beasts.
But they are an exception (the clue us in the name I guess)
Most of your shark alerts at beaches are due to dolphins, who much more regularly stick their fins out of the water. The lifeguard sees a fin and nothing else, and immediately thinks “shark!” and calls the alert. That documentary called “Jaws” shows what happens if you don’t call the alert and there really is a shark. 
Sometimes you can see a shark from a helicopter or plane though. A lot of beaches have planes flying up and down the beach towing banner ads, and sometimes they can spot and track a shark.
This guy, for example, was seen from a helicopter:
He’s pretty close to the surface but his fin isn’t out of the water.
Some beaches have dedicated planes or helicopters (or just boats) for shark patrols. If a lifeguard spots something, they will come and check it out. If they don’t find anything, they’ll re-open the beach.
