Does what descends from a نجسة (ritually impure) hand onto parts of the washing machine render them impure, and also render impure the laundry that touched them? And how is the dryer purified if it becomes impure? And does laundry detergent render clothes impure if it contains impurity?

It is clear that you are afflicted with waswasah (obsessive whispering), so we advise you to ignore it and pay no attention to it. Nothing is ruled ritually impure (najis) except with certain conviction. Mere doubt is not enough to rule something najis. Therefore, your doubt about the purity of the washing powder or the transfer of impurity from your hand to the washing machine should be disregarded. If you become certain of the impurity of a part of the washing machine, it can be purified by pouring water over it.