Would this be to much time? Or should we add a 2nd day on to Dresden or Nuremberg? Thanks.

You could proabably take the train to Dresden for the day and return. This way you only need to deal with one hotel.

Nuremberg is quite a hike. I would stay in Berlin, see Potsdam, Dresden, so that gives you three days in Berlin. If you get bored, check out the Therman spa in Europa Center. Take a bicycle tour through the city.

5 days is probably barely enough for Berlin. It's a big city with lots of sights; esp if you're into art. Lots of excellent museums.

Potsdam is a good day trip. Never been to Dresden, but that would be a great option too - make sure you visit the Dresden Green Vaults.

Look for my recent trip report on Berlin. I had 5.5 days and daytripped to Potsdam. I think Dresden is too far for a day trip, but others have done it.

Definitely not "too much" - if anything, too little! However, as mentioned before, both Potsdam and Dresden are realistic day-trip destinations (there are also coaches leaving Berlin, which might be cheaper than taking a train - see :
http://www.germanytoenjoy.com/indiv/i_dresdentour.html
http://www.germanytoenjoy.com/indiv/i_potsdamtour.html)

There is so much to see in Berlin. Just an example for a 5 day itinerary:
Day I - tour to the major sites + Pergamon (perhaps you can squeeze in another museum? I don't think so)
Day II - Reichstag, Jewish Berlin tour, Jewish Museum
Day III - Wall/Cold War sites, Charlottenburg Palace
Day IV - Third Reich sites, Potsdamer Platz, Kulturforum
Day V - Wannsee, Potsdam, etc.

Notes;
- no dresden here
- so many sites missing, not to mention shopping, going out, etc.

I'm going for 6.5 days at the beginning of July.

Besides day trips, are there places worth exploring by renting a car?

Or even an overnight trip possibly?

The city is big and there seemed to be a lot of sites so I allocated an extra day before moving on to Nice.

Dresden would be my strongest recommendation for an overnight trip by train. Hamburg second, Leipzig third. If you are looking for smaller charming towns as a contrast to Berlin, then head to Quedlinburg in the Harz mountain resort area. It's an amazing town on the hill with a church at the very top. From Quedlinburg you can also visit nearby Wernigerode along with its hilltop castle.