Northern Germany, especially the Baltic Coast and the Mecklenburg Lake District, are a very popular summer destination for Germans, and summer is the perfect time to visit the region. Prices however skyrocket in July and August, but mostly in the spa towns and not so much in the cities. In June and September the weather is still very good, but the prices are a bit lower. Weather-wise May is OK too.
Please note that you only visit a small part of Pomerania on your journey. Most of it is in Poland today, but because of WW2 and the population transfer it's not the same country your family left anyway. German Pomerania is between Stralsund and the Polish border. In Greifswald, a cute, mostly preserved town with a nice harbor, is the Pomeranian State Museum, in case you are interested in the history of Pomerania.
Your itinerary looks good to me. It doesn't make a big difference if you stay in Schwerin or Lübeck. Lübeck is historically more significant, bigger and shaped by the Hanseatic League, while Schwerin is slower, more cozy and basically the 19th century seat of a duke. Each city is good for a day of sightseeing (although some people squeeze Schwerin and Wismar into one day), and on the second day you will visit the other city anyway. They could be done from Hamburg too.
In case you travel in summer I recommend to visit at least one spa town, which make the Baltic coast so popular. Many of them offer wonderful so called "Bäderarchitektur" (resort architecture) from the 19th century:
The most popular resort is Binz on Rügen island, which can be done from Stralsund. Equally popular are the "Three Emperor Spas" on Usedom island, which can be reached by train from Greifswald. If you want to visit a spa town I recommend to spend only 6 days in Berlin. Also consider only 2 days in Hamburg, which IMHO is enough to see the city. This way you could split your time evenly between two big cities and rural Germany. In this case you could also see Bad Doberan, which offers one of the most beautiful churches along the coast, or Wismar, a World Heritage Site, or Ludwigslust, like Schwerin a seat of the Mecklenburg dukes, or Güstrow, a provincial town which probably gives you the best idea how the home of your family once looked.
If you focus on Mecklenburg and Pomerania and spend a week there you could also consider to rent a car. Driving there is easy. The region is sparsely populated, the cities are small and the Autobahn along the coast is the quietest in Germany (the place to drive flat out!). ;) A car gives you much greater flexibility, since train connections aren't always the best, and some places, like the Darss peninsula near Stralsund, aren't accessible by train at all.