i have always used car wash soap,
when i'm doing my full detail procedure then it's dawn dish detergent i use in the bucket of water, otherwise it's regular car wash stuff like such as meguiars gold class. Anything that suds decent will work.
how long the clay lasts depends on the condition of the car, and if you drop the clay. if it's the car's first time being clayed and it's neglected you might need to dispose of the clay afterwards especially if you clay the bottom parts that have tons of road grime, but after that if the car is maintained then the clay can last a really long time. in any event, 1 clay bar should be all you ever need to do the entire car even if it's really bad.
and a good trick is to have a clean small container of hot water, put the clay in it prior to using it so it softens and you can kneed it to expose clean clay.
