Why is the Chlorine missing an electron in Chlorine Monoxide?
In fact the unpaired electron belongs neither to Cl nor to O, but to a molecular orbital spread over both atoms [but not equally: see below].  The Lewis structure is simply not a complete representation of what is going on.  (And for what it's worth wikipedia gives the same Lewis structure you drew: http://en.wikipedia.org/wiki/Chlorine_monoxide)

Edit: Actually it seems the highest occupied molecular orbital (where the unpaired electron is) is mostly localized on the oxygen atom.  

MO Diagram (taken from here):


That is to say, your structure is more correct than the one in the answer key.  However the justification involves molecular orbital arguments outside the scope of a general chemistry course.

Another edit: I too like your argument about formal charges and would accept it as a justification that the unpaired electron belongs to O rather than to Cl.