It seems doable, although at a really frenetic pace. I would check the opening hours of all these places. The Matisse and Chagall museums aren't that far apart, and the Russian Orthodox Church (if I've got the one you mean) isn't that far from them.
The Picasso museum is in Antibes, but you might be able to get down there, and then back to the Rothschild villa (if you're staying in Nice, otherwise the logistics might be different).
Presumably the museum hours are over before dinner, so you can stroll around Vieux Nice and find a restaurant.
Are you going to have a car? That would seem the most practical, but maybe not totally necessary.