Is a young man considered legally excused for not washing his ritually impure clothes himself, and is the house considered impure due to family members using ritually impure clothes within it?

The problem is obsessive whispering (waswasah), and its cure lies in discarding these whispers and not deeming anything impure except with absolute certainty. Your mother's washing of the clothes purifies them if they were ritually impure, and merely pouring water over them is sufficient for their purification. Do not open the door of waswasah upon yourself, for it leads to great evil.