I want others to help answer the Villa access question. But I know this:
In Varenna there are two villas that have their gardens open. They are within walking distance. One is a hotel so the interior is not open at all. At the second, they were doing lots of interior work when we there(The interior was closed and I don't know if the interior is ever open).

From Bellagio, Villa Melzi? is to the south. You have an only 18 inch ribbon of pedestrian area beside all the car and truck traffic but the walk is only a few minutes.

On the other Hand, Villa Carlotta has its own boat dock and is relatively close to both Cadenabbia and Tremezzo.

The two villas I'm interested in (probably only have time for one) are Carlotta and Balbianello.

I'm glad we stayed in Bellagio; we've been there twice and last time stayed for a week for a restful break. My husband is not much of a shopper but still enjoys the town. We walk around, have coffee at outdoor places, and take the boats around the lake. If I were choosing, I'd stay in Bellagio again and hop up to Varenna for a walk/lunch and head back.

If you take the hydrofoil or boat back to Como and then need to get to the train station in Como, be sure to get good directions about where to catch a taxi to the station.