Does impurity spread from a wet hand by touching the wall and the faucet while washing hands after purification (Istinja'), and are the prayers of family members invalid if impurity transferred to them and I did not inform them, and am I accountable for that?

If urine soils the hand, running water over it purifies it, and the wall does not become impure if it is touched after water has run over the hand.

If you are certain that impurity has spread to a specific spot, it must be purified with water; otherwise, it is a whisper from Satan (waswasah).

As for the family's prayers, they are valid if they were unaware of the impurity's presence, because avoiding impurity is a condition only when one has knowledge and ability.