For stop 2, besides the reasonable options that have been discussed, the city of Burlington operates a campground at North Beach park. The beach on the lake is very nice and sandy and often popular during the day.
I woukd think twice before hiking up Mt. Washington. There are no lack of slightly less challenging but still impressive mountains to hike. Camel's Hump is one lovely option as an example. The cog railway or auto road (if your vehicles would be permitted) are good options for Mt. Washington.
It is something like a 3.5 hour drive from Burlington to Mt. Washington, probably another hour and a half from Lake George. It would not be practical to do that as a day trip, in my opinion.
