We just took a very similar trip last month, although we flew round trip from Newark, NJ. We used 60,000 United miles per ticket for 4 of us (240,000 miles total) and were able to fly the following journey:

FLIGHT INFORMATION

Wed, xxJUL13 UA8 XN NEWARK, NJ
(EWR - LIBERTY) 9:00 PM LONDON, ENGLAND
(LHR - HEATHROW) 9:25 AM (04JUL) 757-200 Dinner

Sun, xxJUL13 OS452 X LONDON, ENGLAND
(LHR - HEATHROW) 9:15 AM VIENNA, AUSTRIA
(VIE) 12:40 PM A-320 Snack
Flight operated by TYROLEAN AIRLINES FOR LUFTHANSA.

Mon, xxJUL13 OS731 X VIENNA, AUSTRIA
(VIE) 12:50 PM DUBROVNIK HR
(DBV) 2:10 PM A-321 Snack
Flight operated by TYROLEAN AIRLINES FOR LUFTHANSA.

Sat, xxJUL13 OU492 X SPLIT HR
(SPU) 2:10 PM LONDON, ENGLAND
(LHR - HEATHROW) 3:50 PM A-320
Flight operated by CROATIA AIRLINES.

Sat, xxJUL13 UA115 XN LONDON, ENGLAND
(LHR - HEATHROW) 6:00 PM NEWARK, NJ
(EWR - LIBERTY) 9:20 PM 757-200 Dinner

We actually considered staying in London longer and taking a train to Paris for a few days, but opted not to do so. We had one stopover (London) and one open jaw (into Dubrovnik and out of Split, Croatia). We also stopped in Vienna for 24 hours (intentionally) since there was only one flight a day from Vienna to DBV and it left at about the same time as the London flight arrived, so there was no way to make the connection same day. this allowed us to tour Vienna for a day (and the airport is only 15 minutes from downtown Vienna).

Lastly, I would HIGHLY recommend the Radisson Blu Hotel Dubrovnik Sun Gardens. It is a truly stunning resort, only a few years old, and can be entirely free on points. We signed up for the Club Carlson credit card, which with the spend was enough for 3 free nights. Additionally, for 70 euros per night, they upgraded us to a 2BR condo with full kitchen, and it included breakfast for the 4 of us. We checked out a bunch of the higher end places in DBV, and this was the best by a long shot.

Hope this helps.