Yes you can have both an H1 and an EAD. I did for a couple of months until my H1 expired; EAD is still valid. If you use your EAD to work however, you loose your H1 status, so it may be best to use EAD as just a backup unless you have no other choice.
An AP is needed if you are working on an EAD or if your passprt's visa stamp has expired and you don't want to get it renwed abroad (there is a procedure for renewing it by mail in the US, but I'm not sure of the details). If you have an unexpired visa in your passport and your status is H1, you don't need AP to travel abroad.
rgd,
sadiq