We want to spend three or more months in Europe and consider Annecy, France a good choice as a base, because of its own assets as well as its convenience to many other places in central and Western Europe. So, considering weather, crowds, cost- would it be best to do this in April-May-June, or August- September- October?

Weather is always unpredictable in this part of Europe. Spring may be a bit more rainy, but days are longer.
August is the main holiday month in France (mid July until end August), there will be much more people in town, but I wouldn't speak about crowds.

All depends on the places you plan to visit from Annecy. For hikes or running cableways in the Alps, August and the first half of September will be better than May-June.
For running alpine infrastructure in October, you will have to go to Switzerland.

Thanks. You correctly note the dilemma of Spring rain v. shorter Oct. days. Not a real "problem" as problems go - after all, we'll be in a beautiful part of the world- but just want to maximize weather and minimize travel issues (to Germany, Switzerland, northern Italy and other parts of France).

Are you a US Citizen? There is a 90 day limit as of now; just in case you are and didn't know it.

We have family that lives in Annecy and the summer can be quite humid and sometimes stormy, but as others have said it will really depend on the kind of activities that interest you. It is a lovely area.

I'm aware of the 90-day restriction generally, but believe it applies only to France and not the EU. Am I right? In other words , I think we could travel beyond France if we want to extend our stay w/out running afoul of the law.

And yes - US citizen.

No, you are not correct. It is for most of the EU.

http://www.projectvisa.com/includes/schengen.html