Update on the internet problem: We've got a solution that we're going to try soon. First, we'll plug the telephone (with a filter) into the current phone jack that the DSL modem is plugged into, so we can use the living room phone. For the DSL modem, we've got a phone jack in our hallway that's unused. Since the hallway (and the phone jack) is not that far away from the modem, we're going to get a long telephone cable, plug it into the surge protector (we have the modem surge protected), and then plug the other end into the hallway phone jack (directly into it, NOT with a filter). We're not going to have the wire stretched across the floor, though, as that would be a bad idea, because if we do that our cat(s) will probably chew on the wire, and we don't want that to happen.
Instead, we'll just wire it off to the side (along the wall), and we'll wire out around the door frames, and then plug it in to the jack in the hallway. That way, the cats can't (or have a very hard time trying) chew on the wire, and the connection problem will be fixed because the filter can't cause an interference with the modem because they're on 2 different jacks.