There are lots of reasons that you might not be able to connect to the Internet. The very first step of troubleshooting isn't discovering what it is, but where it is.
Is the modem loosing sync? Or does the modem remain online while the computer is unable to connect? That's the big question right now. The modem should be able to stay online even if there is no computer connected to it, or even if there is a broken computer connected to it. Think of it like dial-tone on the telephone. It doesn't matter if the person trying to make the call can speak English or not. It doesn't even matter if it's a person picking up the phone. If there's no dial-tone, a call can't be made. Teaching the person to speak English isn't going to help.
If your new SurfBoard modem has 3 or 4 solid green indicator lights when you loose connection, then the modem is not loosing sync. If it's cycling through a sequence of flashing lights, or if any of the green lights are flashing, it is probably loosing sync.
If your modem is loosing sync, then Adelphia is going to have to figure out why. It could be the cabling in your house. It could be the cabling anywhere between your house and their headend. But it's someplace before that cable screws into the back of the modem.
If the modem is not loosing sync, and your neighbors aren't also having problems, then it may be a problem with your computer. I highly doubt that it's the motherboard. I highly doubt that it would be anything related to hardware.
Whether the problem is in their network or your computer, the issue may take quite a bit of interactive troubleshooting to isolate it and fix it. but before anything more is done, it needs to be determined on which side of the modem the problem exists. Which side of the modem the problem exists on makes a major differerence in how the problem is addressed.
The 4 steps you've said have already been taken indicate that either people who should know what they're doing are idiots or simply don't care, or possibly that you had multiple problems. Perhaps there was a problem before the modem that has now been fixed, and they now can see there's another problem with your computer. But if there has only been one problem all the while, then they're missing that very important first step of identifying on which side of the modem the problem exists.