It depends on what you want to do after arriving in those places. For example:
Menton has a couple of nice gardens quite some distance from the center of town, though one is walkable from the second train station. There's also the picturesque (if very touristy) historic center and at least one small art museum.
St. Paul-de-Vence has the impressive Fondation Maeght, a bit of a walk from the village itself.
Nice has at least six art museums as well as other sights.
Antibes has a Picasso Museum.
Cap Ferrat has the Villa Ephrussi de Rothschild, with very nice gardens, and the Greek-style Villa Kerylos is located just off the peninsula, in Beaulieu-sur-Mer.
Those are just examples.
I'd take into consideration not only my level of interest in the specific sights in each place, but how much transportation time would be required to reach them. St-Tropez is more challenging than the other coastal destinations because it has no train station; you'd need to use a ferry, bus, rental car or taxi for the last part of the trip. Getting to SPdV involves a bus, and I think it's no longer just a single bus from Nice. You also need a bus ride (albeit a shorter one) to get up to Eze Village. In contrast, Juan-les-Pins is walkable from Antibes, and Antibes is on the easy-to-use regional train line.
I opted to stay in Nice; that made it easier to get to all the city's art museums. Others like a quieter environment and often mention Villefranche-sur-Mer. Consider which of the towns you've listed are your top priorities and choose a place reasonably convenient for those.
I wouldn't be able to go to all the places you've listed in a week. I'd start by estimating how many days I'd need for the sights in Nice alone.