Just a couple of things that may clear up some of the mystery about the hardware showing on the machine. The Agere modem is the builtin dialup modem and will always show as hardware installed-not actually the previous owners modem..other than she may have used it for dialup.You can go into Internet Options in Control Panel and on the Connections tab see if any dialup info is there.If so click to highlight it and then click the Remove button.But the Agere modem will always show in Device Manager because it is internal hardware. The Realtek you see is the internal network card that your modem plugs into via the orange cord. It is the only hardware that will show associated with your internet/modem. Your modem is an external device and won't show up as hardware. (The machine also has Realtek AC97 Audio which will show as installed hardware)
Hope this helps with some of the hardware questions. As far as the software problems, IE, Firefox,etc I think you may just have a very fouled XP install which has likely never been refreshed with a Recovery. An operating system does require a reinstallation occasionally to keep it running well.