I had the time to do a few quick tests of flak vs aircraft and I have to take back a lot of what I said here.
Flak is effective - it is not a magic cure but a reasonable amount of flak goes a long way in protecting your forces from air attacks. And yes, planes can get shot down.
I took the same map we have with Stafford, placed 10 StuGs roughly where I placed them in the battle, and deployed an AA company in the setup zone (~600-800 meters back) with three platoons: 3x 20mm single, 3x 20mm quad, 3x 37mm. I tested each platoon alone in each run, ordering the others to hide. Therefore, no more than three guns fired in each test.
Four Typhoons attacked in preplanned strikes, two with a 5 min delay, one with 10 min, and the last one with 15 min. I spread out the attacks so that all StuGs could be targeted. I only did each test once.
First run - no flak. All planes attack until they run out of ammo. 5 vehicles destroyed, 4 disabled, 1 undamaged
Second run - 20mm singles. The planes spent on average only about 40% of their ammo and then broke off the attack. Some broke off immediately and not fire at all, some attacked with full effect, but on average the damage was much less: 4 vehicles disabled, one lightly damaged.
Third run - 20mm quads. In this case I didn't seem to be substantially disrupting the attacks, but one plane was shot down. The rest spent all their ammo, destroying 4 vehicles and disabling 1.
Fourth run - 37mm. The planes spent about 40% ammo on average and one was shot down. 5 vehicles disabled.
I think that in my situation, a platoon of light or medium flak (something like 170-240 points) could have reduced my casualties by about a half. The problem remains that even after the planes break off the attack, they remain available to your opponent and can be called back. Meanwhile, the flak runs out of ammo rather fast - after the fourth plane the 20mm quads and 37mm were generally Empty. The 20mm singles still had a decent amount of ammo left, which makes them the best choice in my opinion - though they didn't get any kills.
For a large battle like this mobile flak would be preferred, but this would eat into the 30% point limit - so I don't think I will accept this limit next time.
When a plane is shot down, it appears as "KNOCKED OUT" on the opponent's air panel. In the next turn, the plane crashes somewhere on the map with the impact of a bomb, and the label "Aircraft Destroyed" appears above the gun that is credited with the kill.
The battle would be more interesting had Stafford not brought so many planes. It would also be more interesting had I brought some flak. A limitation of the maximum number of planes is still reasonable (e.g. I wouldn't want to face 20 planes attacking at one time), but the German player can protect himself - it's just my problem that I didn't.