You might not have heard about Card Case. It's not the most popular of Square's innovative services, but it's worth just as much attention for the average consumer. Now that it's getting a facelift, it's a good time to revisit the concept. Pay With Square (as it's now known) is an app that allows you to create and maintain tabs at local businesses by simply giving the retailer your name.
To answer your first question, no there's not a lot of places that support this payment method just yet. However, the app does have a built-in directory of the places that do support it, which is handy for new payment methods.