This video was shown to incoming US bomber crews to show them why they used certain tactics to avoid anti-aircraft fire. You can also watch it to learn how antiaircraft fire worked at the time.

Most rounds were indeed designed to explode at a predetermined altitude and use their shrapnel to damage the planes they were targeting. I believe--but can't be sure because the sources I have ready access to don't mention it either way--that most antiaircraft rounds were also fitted with an impact fuse for the rare instance of a direct hit. (Edit: Pgs 255-257 of this declassified USAF document (PDF warning) asserts that the "double fuse" of a preset altitude and impact fuse was discontinued by the Germans early in the war to avoid premature detonations on the ground, and was only reintroduced in 1945.) As to how the fuses were changed, that was part of the process in loading and firing the gun. One had to determine the altitude and airspeed of the plane you wanted to attack, then calculate what fuse or fuse setting should be used to achieve your goal, then fire the round at the right trajectory to intercept your target.

It would be much easier if you could just fire the gun in the path of the plane, and have the round itself sense when it was near an airplane. These were called Proximity Fuses or VT fuses (for Variable Timing). Early in the war the Germans ceased research on this type of fuse, but the Americans and British combined their efforts and were able to construct such a device. When affixed to antiaircraft shells, this combination was devastatingly effective. In fact, its usage was restricted to where Axis forces could not recover a dud round--over friendly territory or over water--for much of the war.

Watch the video, though. The decision loop for firing at intruding aircraft is fascinating.

Disclaimer: Not an expert but a keen student of WWII history.

Now that's out of the way we can crack on.

I'll start with how they were meant to damage aircraft. You're on the money with shrapnel. Given how hard it is to hit a moving plane at 20-30,000 feet trying to get close enough to hit it with concussive force would be too difficult. Shrapnel rounds would cover a large area and have a higher chance of hitting something. If close enough they would shred aircraft to bits, either knocking out vital systems or tearing structures apart.

Fusing. For the most part simple fuses were timers, you set it to explode at a certain height for larger artillery pieces. This is known as air bursting. The AA crew is told that the plane is at 25,000 feet so they set the timer to give enough flying time to get to 25,000 feet and then explode. A lot more complicated but much more effective were proximity fuses. As the shell passes the aircraft the metal of the aircraft triggers a magnetic switch, completes the circuit and boom.

For smaller cannons, and aircraft cannons I'm sure that impact fuses were more common. It hits something, it goes boom.

Any experts can feel free to correct me but that's what I remember.