When I was using a modem on a regular POTS line, a caller would get a busy signal while I was on the Internet. Just like when you have a conversation. If you try to call out while on the Internet, you will disconnect the modem/Internet. It works exactly as if two people wanted to talk at the same time on the phone, you just can't do it. At least that was, and I imagine still is. Your best solution is to get another POTS line. One for the modem, one for voice.
As far as Fax/Modem, it works just the same way, just don't add the fax portion if you don't intend to use it. I think this is more for marketing then anything. It sounds like you get more for your money by saying Fax/modem over just modem. You might get more money for it too.
So don't be afraid if it says Fax/modem... I would prefer making sure it was full duplex.
Good luck!