I would start with $100-$200 and then get more as needed. Some of the food carts only take cash. I personally like to travel with more cash than I think I will need, in case of some bizarre scenario that may come up.
I don't know who your bank is, but there are Chase ATMs in various locations throughout WDW, including inside the parks.
I suspect, though don't know for sure, that you could select the cash back option when using a debit card during a transaction at a store.
I have never paid for parking (always stayed on site), so I don't know what forms of payment are accepted.