If you are getting a car ready to clay/polish/wax/whatever then dish soap is great as it will remove any wax that is currently on the car. After you wax it though, its best to use a car specific wash formula so it doesn't take the wax off.
I much prefer the paste was actually. Its a lot easier and more precise to apply it than liquid wax, plus the spongy applicator you mention works great. Just make sure you keep the plastic thingy that seperates it from the wax and store it just like it is when its new. A friend of mine has a tin of it and he left the sponge sitting on the wax and the sponge is all nasty looking now and the wax is cracked and dried out. Looks to me like the sponge sucked the moisture out of the paste while it was sitting there and dried it out.
I usually let my cars air dry, but I know I have good water that won't leave spots. You can get a big microfiber drying towel. There are a few different types, the waffleweave ones I've heard work the best, but I dunno. Use the regular ones for wiping the quick detailer you use with the c;ay and for buffing the wax off.