Do any of the email's get returned to you or do they just go missing? If they get returned there should be a reason.
Try plugging your IP address into the BlackList tab to see of it has been flagged for spam.
When you say it will not "hang up on its own" I assume you must do something to indicate the call is over, what is that. Hard to imagine what could be unique about the FairPoint network to prevent the phone from operating properly. Hanging up is simply a matter of opening the circuit, so loop current goes to 0, normal loop current is about 20-50ma. When this occurs how do you end the call?
When did the problem start? Is this a new device that has never worked correctly on this line or did it used to work and now has failed?
Is it possible to have someone try the phone at a different location, to see if the problem travels with the phone.
Is it possible to perform a test with a regular phone and see of it experiences the same anomaly. If so my guess is there is some sort of marginal short on your line that is able to conduct enough current to prevent the switching system from detecting the phone has been hung up.