To fiat that'll take a while, but moving it around the space cash world is often very quick, on GDAX anyways.

I do it for a living.

How do you query the card balance and transfer the funds? Do you actually get cash out of the card, or do you have to resell it as a store credit to someone else?

Long story, and it very much depends on the brand. Re: cashing out, we do both transfers and resale, depending on the user risk profile.