Down-filled means the pillows are filled with goose feathers, which makes the pillows fluffy and soft. If you have gotten regular pillows, they're likely filled with polyfoam, the same cottony-fluffy looking synthetic stuff that is used to fill toys. You can put those in the washer. Just balance out the load with two pillows like I suggested or else your washer will be skooting all over the floor and making a huge thumping noise and scaring everybody

There may be some pillowcases available that goes underneath the cloth pillowcase - something plastic, or if you're in a pinch, you can always use bathroom sized garbage bags (small), those should be the same size. You can probably go to a specialty store that sells bedding for kids. I'm sure they have special bedding for bed-wetting children and perhaps they have covers and padding that repels stains and drool.