APA doesn't clarify whether the cue ball has to be motionless prior to the breaking stroke. You can adjust the position of the cue ball with the pool cue prior to breaking. The only stipulations on fouls while breaking is the difference between a legal break and an illegal break.
legal break:
cue ball struck from inside the kitchen
strike the rack prior to striking a rail
4 object balls driven to rails or object ball pocketed
the double hit on the break question hits one of two situations:
1) you double hit prior to breaking the rack, so the cue ball is in motion for the breaking stroke
2) you hit the cue ball with your stick after breaking the rack.
case 1, if the cue ball is still in the kitchen the second time you hit it, it's a legal break in the APA. If it was outside the kitchen on the second hit, it's an illegal break. if you didn't scratch, your opponent reracks and you break again. no foul. if you scratched, you rerack and your opponent has the break.
case 2, you fouled the cue ball after striking the rack. If it was otherwise a legal break, you opponent has BIH from the kitchen. If it was otherwise an illegal break, you rerack and your opponent breaks.
That's the APA perspective, anyway. I wasn't sure what kind of double hit y'all were talking about, so I figured I'd cover all the bases.