What is the ruling on someone who finds a substance preventing ablution on the sole of their foot after prayer and does not know when it got there? Should they repeat the prayers from the last ablution, or from the beginning of the day, or from yesterday? And what is the permissible amount of a substance preventing ablution that is overlooked for the sake of ease?

Ablution (Wudu') is not valid if there is a barrier preventing water from reaching the skin. If such a barrier is found and its duration is known, the prayers performed during that time must be repeated. If its duration is unknown, then your last ablution and prayer are valid, and the presence of the barrier is attributed to the most recent possible past time. Then, the barrier should be removed, and ablution performed for future prayers. There is no exemption for a barrier that prevents water from reaching the skin, even if it is slight, contrary to the view of some jurists.