The overhaul kit just includes what you need for installing the pistons back into the calipers.
47389-24010 Gasket for flexible hose
You'll need two gaskets for each caliper, I don't know how many you'll get in a pack at Toyota or if you need to order eight times the part number above.
No problem pushing the car with calipers removed, I sometimes just put a brick or a chock a few feet infront of the wheel so it doesn't move too far and damage something.
I don't think the Supra has screws holding the brake discs on the hub, so when you remove the caliper/pads there'll be nothing holding the discs in place until you put the wheels back on apart from maybe a little rust.