How can one determine whether the dirt on the body – especially around the toenails – is from dried sweat or dead skin, and must it be removed before performing wudu'?

It appears that the hardened sweat around the toenails does not harm or invalidate the ablution (wudu’). As for dead skin, it is a barrier that prevents the validity of ablution, because it is a dry part of the skin, and if removed, the person would not feel pain. However, any skin or peel that cannot be removed without causing pain does not need to be removed, and purification is valid with its presence.