Should my a/c have some sort of a drip pan (photo)
Yep, you do have a built-in drip pan. However no, you should not install any additional drip pan. That "radiator" (evaporator coil or A-coil, due to shape) is what the air travels directly through in order to remove the heat from the house when cooling is in use and what heated air passes through during heating season...any additional air restriction would be a bad thing.

I'd say be extremely careful in cleaning, as to not bend any of those fins. But, you really should call a Heating and Cooling Company or Guy and have them thoroughly inspect, repair, service and clean the entire system. Yours looks overdue and a pro's couple or few hundred dollars will come right back to you in restored energy efficiency for both heating and cooling.