What is the ruling on the residue of soap or ink remaining on the hands during ablution, and does this invalidate ablution, and does the mixing of water designated for ablution with soap or other substances spoil ablution?

If what is on the hand has a substance that prevents water from reaching the skin, it must be removed before ablution. However, if it has no substance, it does not prevent the validity of purification.

As for water mixing with soap: if the water changes as a result, ceasing to be pure water (mutlaq), purification with it is not valid. If it does not change, or the change is minor, then purification is valid. Al-Zarkashi mentioned the difference of opinion among scholars on this matter in his commentary on Al-Khiraqi, indicating that if water significantly changes in color, taste, or smell such that it becomes attributed to something else (like saffron water), some scholars believe its purity is lost, while others hold that it remains pure as long as it does not lose the designation of pure water (al-ma' al-mutlaq).