Resident IT guys, read the posts - we're going way over his head.
First off, you don't need the phone company's switch config. I manage multi-site networks with both voice and several data segments, and have never needed a provider's router config beyond the customer interface (which I specify to them). Since you're only experiencing trouble with some phones, you're past the point of worrying about that anyway.
Who configured the phones? What are the phones connecting to? Is the Adtran part of the phone picture, acting as a service router / SIP tunnel?
Sounds like this is a very small office, and some phones - those before your HP switch - work fine. You may just want to turn that HP switch into a "dumb" switch by setting all of the ports to be trunks. Sure, there's caveats, but it likely won't matter one bit. If you take a non-working phone and plug it in place of a working phone, does it too work?