1. - Normally the phrase is "over-the-counter medicine," which means that you did not need a doctor's permission to get the medicine. "On-counter medicine" sounds like when you went to purchase the medicine you walked up to the counter and the medicine was available for purchase literally being displayed on the counter. It does seem to imply that you did not need a doctor's permission, though.
2- Yes you can use "clear up" to describe something which is not clear, or to describe whatever is making something not clear. In this case the sky is not "clear" because of the clouds or cloud system. Then the cloud system "clears up".