Is the prayer valid after masturbating, then washing the penis and testicles and purifying and sprinkling the garment, then performing ghusl and wearing the same garment for prayer? And is it permissible to touch the penis wet with semen and pre-seminal fluid to wash it from urine, and is it obligatory to wash the hands thereafter?

You must repent from the bad habit. The prayer is valid because, according to some jurists, sprinkling water (nadh) on the garment is sufficient for purifying it from madhi (pre-seminal fluid). However, it is more cautious to wash it to avoid disagreement. There is no harm in using the hand when performing istinja' (cleaning oneself after relieving oneself) and its touching impurity while removing it.