I didn't wash the cotton mitt before I used it. Someone also asked about the DFtowels I used. They are amazing. I just emailed them to see if they make a wash mitt.
Also, running water, if available, is safer than the two bucket method if possible. Run the water through the mitt as you wash. This floods the surface and keeps debris away from the paint.
Without soap. I only use soap every three washes or so. My cars don't normally have a chance to get "really" dirty so I refrain from too much soap which is sometimes kind of harsh on wax longevity.