Hi mariabjorgensen!
Having in mind your itinerary and your wishes concerning the activities and the proximity to the airport, I can share with you the following thoughts.
Honestly it is pretty hard to find a place close to Dubrovnik airport and to fulfill your other requirements. There are several good offers in the villages close to the airport like Čilipi or Zvekovica, anyway I seriously doubt that they would be suitable for such a young couple and beside this, they are not close to the sea.
So, in my opinion, the best option for you in Dubrovnik would be some nice apartment which is close to the old part of the town and with several amenities like shops with foodstuffs (so that you don't have to spend too much money in the restaurants). When we talk exactly about the accommodation, then my favorite is Apartments Stella. In my opinion, the best option is to book a double bed private through Hostelworld. It is also importatn that this accommodation is close to the local bus station, so you can use it to reach other parts of the town, especially the bus station and seaport.
As far as the itinerary is concerned, I would suggest you that you spend one whole day for the island of Lopud. For example, you can take a cheap, public boat from Dubrovnik at let's say 09:00 or 10:00 in the morning (departure depends on the day of the week), then explore and enjoy this wonderful island and then return by boat at 18:55 or 19:45 (again, depending of the day in the week). Here is the timetable: http://www.jadrolinija.hr/pdfs/807.ljeto.2013.e.pdf and please, look at the second page of it. Also, you should definitely visit one other, but far smaller island close to Dubrovnik. It is Lokrum, but although it is small it has a lot to offer to its visitor. It is connected by local small boats from the port situated in the old part of Dubrovnik.
Now, as far as your part of the itinerary for Montenegro is concerned, just be informed about direct bus line from Dubrovnik to Kotor, because they are rare and their schedule is pretty erratic. My suggestion is, if you know the exact day when you plan to travel to Kotor, go to Dubrovnik bus station, look for the schedule and buy the tickets. So, you will have a peace of mind.
My suggestion for your planned days in Kotor are that you find yourselves a good accommodation and to explore the first the gems like Kotor itself and then Perast and Prčanj also. Perast is the true gem and in front of it there are two small islands with the monasteries. Kotor is also a good base for trekking, so you can enjoy also in this part of its offer. Of course, if your time will allow you, I definitely suggest the visits to Lovćen national park and to Cetinje, a small town that used to be the historical capital of Montenegro.
For the remaining three days I definitely suggest that you visit Mljet national park:
http://www.youtube.com/watch?v=2JcqrPy3wR4
This is truly amazing place that allows you to enjoy the nature in full. You can enjoy kayaking, trekking, cycling... There are so many activities on your disposal. It is very easy to reach Mljet national park by catamaran from Dubrovnik. Take a catamaran from Dubrovnik at 09:15 daily to Polače where it arrives at 10:50. Find a suitable accommodation and start to enjoy your time. You will enjoy every moment of your sojourn there. To return to Dubrovnik, just take a catamaran from Polače at 16:55 daily, with arrival in Dubrovnik at 18:35. Here is the timetable: http://www.gv-line.hr/raspprint.php?linija=3&red=62 . The price for a single journey is 54 HRK or roughly 7.2 EUR.
I would like to inform you about one more thing and it is up to you to decide. Well, I always recommend to the people to use the public transport from the airport to Dubrovnik, but in most cases, people tend to use taxis. In the case you use the taxi, please, keep in mind that it is far more expensive than the public transport and the price should never exceed 30 EUR. Sometimes the taxis try to even more inflate the price, so keep this data somewhere in your mind.
Enjoy your holidays.