Do tiny, hard-to-see black dots on the foot act as a barrier to ablution, preventing water from reaching the skin, and are ablution and prayer valid with them?

What you have mentioned is merely a whisper from Satan (waswasa), so do not pay attention to anything you have mentioned, and do not repeat your prayer no matter how much Satan whispers to you that it is invalid. Doubt should not be given attention, especially if it occurs after performing the act of worship. The one experiencing these whispers should turn away from them and not give them any importance, as dwelling on them leads to great evil. Even if it were assumed that there was a small barrier present in reality, not merely a doubt or waswasa, then purification is still valid with it according to Sheikh al-Islam Ibn Taymiyyah and those who agreed with him. The one suffering from waswasah may adopt the easiest of opinions to alleviate hardship.