My Walther P22 works just fine with Federal American Eagle bulk .22, but the instruction manual that came with mine said "Don't consider the gun broken in until at least 500 rounds."
The Winchester white box bulk ammo is iffy for me, maybe 80% reliable. Almost no "standard velocity" round is 100% reliable.
As I mentioned in another post, with some ammo, grip can play a pretty heavy role in reliability for the P22. The hotter the round, the more reliable it is. The CCI MiniMag has been the most reliable, but for my Walther P22 it's not the only thing that works for me. Federal bulk ammo works like a charm as long as I make sure I don't limp-wrist the shot.
As far as the "second round" issue goes, I haven't had any problem like that with the three magazines I have for the Walther. I have had "too full" magazine syndrome on my GSG-5 where the second round fails to feed if I put 23 rounds into a 22 round magazine. There is a chance the magazine is to blame. Try loading 8 rounds and see if it behaves. Then increase to see if it hits a problem.