No, you can't use the card while online. The shopping card can only be used at Gracie Grace's and be used to hold money. And when you're at a friends town online, you can't go downtown, and downtown is the place you need to use the card. So no to both questions.
Is it like a real credit card where you have to eventually pay it off or is it just a very large money supply that you can blow on anything you want without consequences?