Trying to get to Dubrovnik June 1-7.
1) Do I have enough time to do a few quick apps to get there for "free"?
It wouldn't be unreasonable to think that you could pull off an app-o-rama to get the needed miles, but you'd want to spend very quickly to have the best possible availability.
3) Most viable options as far as routing and airlines?
It seems like most of the airlines that fly into Dubrovnik are seasonal. Is there another nearby airport you could fly into? Would you be comfortable flying out of another airport in Florida besides JAX? As always, with more flexibility comes more options.
DL doesn't book tickets online for DBV, so that's out. BA flies from LGW to DBV on a regularly-scheduled flight, but I don't see any availability for two weeks around your stay. UA has award availability from JAX-CLT-FRA-DBV on the outbound for 30k UA points, but that's using US Air flights that are about to disappear and a Croatian flight from FRA-DBV that's about to produce an increase in price, I believe. They also don't have outbound availability.
I think your best bet is probably going to be a pair of reservations. EasyJet serves Dubrovnik from a number of European destinations, including London, Milan, and Rome. You might want to get a pair of cards that you can pool together -- I would figure one of the Chase Ink/SP cards and the United card -- and then book a TATL trip using the miles from those cards before an EasyJet trip from one of those European destinations to/from DBV.