Almost all night life is concentrated in or around Old Town Dubrovnik. Also, all mayor monuments and other cultural sights are inside the City Walls, so one option for you would be to find accommodation in Old Town Dubrovnik.
Other areas, such as Lapad peninsula are also very nice and there is a regular bus service every 15 minutes to and from Old Town.
For acommodation you can try http://www.dubrovnik-online.net/english/private_accommodation.php, there are many rental properties in and around Old Town. about 10 years ago
I agree. You should stay in or close to the Old Town. Lapad is geared more toward families and resort life. We stayed in the Sesame Inn last summer, it's a short walk from the city walls so you can sleep quietly yet are in the midst of everything. http://www.sesame.hr/inn.html
They have a delicious restaurant as well.
You could also consider Budva in Montenegro, which is also a very ancient city but a lot less over-touristed. Dubrovnik can get a little crazy.
It's not too far of a drive from Dubrovnik. My father is from Montenegro and I love that little country :) about 10 years ago