Fayence is much closer to the Riviera than to the most-mentioned places in Provence. Even Aix is almost a two-hour drive from Fayence. Getting to the Luberon would take a bit more time, and Arles and Avignon are even farther away. I suspect your villa group won't actually go to Provence, so you might want to spend a good chunk of your time there. Honestly, I'd make it more than 3 nights if I were going to head in that direction; I'd hope to see both Arles and Avignon as well as at least one smaller town. Avignon seems a bit better than Arles as a base for those without a car, but if Arles sounds more appealing to you, it will work. Aix is nice, but the location isn't quite as convenient as a base for side trips for carless travelers.
There's a disadvantage to heading west from Fayence: The train trip back to Nice isn't particularly fast, so that will cut into your week of sightseeing time.
I haven't been to all the places you mentioned, but I have spent a couple of weeks in Nice (May 2017) and day-tripped to Menton, Antibes, St-Paul-de-Vence, Vence and Cap Ferrat. I didn't find those places to be repetitive. Vence is the least touristy of the group; it's easy to combine it with St-Paul-de-Vence by bus from Nice. Cap Ferrat is notable for being the location of the Villa Ephrussi de Rothschild (with nice gardens) and the Greek-style Villa Kerylos. Menton also has some nice gardens (not right in the middle of town).
St Tropez is more complicated and I wouldn't bother with it while staying in Nice since there are so many other, more convenient, options. St-Tropez doesn't have a train station, so it requires a ferry or a bus (or maybe it's a combo--I haven't been there).
There are a couple of big-deal events along the Riviera in May, the Cannes Film Festival and the Monte Carlo Grand Prix. I was told it's the Grand Prix that most affects lodging rates in Nice. You might want to go ahead and book a cancellable room in Nice for your week, just in case you decide to head in that direction. Rooms are highly unlikely to become cheaper between now and May.
The surge in visitors for the Riviera special events can lead to lines at the train-ticket vending machines in the Nice-Ville station. There's no need to buy the tickets early for trips along the Riviera (fares don't change and there are no reserved seats), but you don't want to miss a train while you're in line at a ticket machine. I'd download the SNCF app so you can buy tickets that way. If you don't want to deal with an app, buy your ticket the night before.
The train along the Riviera is quite inexpensive and is popular with pickpockets--probably especially during the film festival and the Grand Prix. Be careful with your purse. Assume anyone who gets into your personal space needlessly is going to try to steal something.