"[1] The MAC address was "unique" in as much as a phone number is. "
Theoretically this is true, but it is possible to spoof a different MAC. But in any case, as Chuck says, the MAC would not normally be exposed to the internet
"[2] the "card" is built into the driver, hence a new driver install"
The card is a physical device, the driver is the software that interfaces it to the operating system. The MAC is normally programmed into the card during manufacture, and is unaffected by changes to the driver (except in some cases where the driver allows you to override the "true" address, and spoof a different one - but even then, the MAC would not be tied to the driver itself, i.e. changing the driver would not automatically change the MAC).
"[3] We don't want or need a wireless router at the moment, so do we need a cable router? wouldn't this be something the ISP would supply?"
Some ISPs supply a modem, some a modem/router, but that's like a "starter kit" - you are not tied to the equipment supplied by the ISP (any more than you are tied to telephone receivers supplied by your phone company)