You didn't say what size tires you already have. So I looked:

Specification Sidewall Radius Diameter Circumference Revs/Mile Difference
235/65-17 6.0in 14.5in 29.0in 91.2in 695 0.0%
235/60-18 5.6in 14.6in 29.1in 91.4in 693 0.3%
235/55-19 5.1in 14.6in 29.2in 91.7in 691 0.5%
235/50-20 4.6in 14.6in 29.3in 91.9in 689 0.8%

I don't know why you want to go with 20 inch, except for the harsher ride due to the smaller side wall. Increasing radio noise to compensate for noisy tires is a poor excuse in my world, I like to hear what is wrong with my car or other cars approaching from behind, like cops and ambulances.
Bridgestone makes a very good all season that is also quiet.

As for getting 4 winter tires, sounds like a good plan. But, do not get wide tires that ride on top of the snow and ice, get something narrower like 205.
--
My back feels better when I sit in a Volvo seat