Would I be considered a half time student if I started the fall semester with 9 units but dropped a class and ended with 4 units? I’m assuming not.

You must have at least the number of units during the semester for being considered half-time by the school you are attending.  If 4 units for the semester is less than half time for your school then you would not be eligible for the American Opportunity Credit.

Would I be considered a half time student

You will have to ask the school. The IRS rules do not provide a definitive definition for a half time or full time student. What the IRS rules in IRS Publication 970 to day is, each school has their own definition of a half time student.So you have to go by your school's definition.

Yes. The fact that you were more than half time, for some part of the term, makes the expenses you paid eligible for the education credit.  If that money was refunded, you not eligible