The ring set at the link below should work.
https://www.opticsplanet.com/ruger-m77-scope-ring-set-30mm-high-blue-90407ru.html
You will need a #4 Ring for the rear ring. The Ruger Scope Ring 4K30 Medium Stainless Steel 30mm model can be found at the link below.
https://www.opticsplanet.com/ruger-scope-ring-4k30-medium-stainless-steel-30mm-90285.html
For a Ruger M77 with a 56mm objective lens, I would recommend using extra high scope rings to allow for clearance of your objective lens. These rings will only clear a 52mm objective. I'd recommend the Super High Leupold Ruger Rifle Scope Mount Rings, found by using the link below:
https://www.opticsplanet.com/leupold-ruger-rings.html?_iv_ring-height=super-high&_iv_tube-diameter=30-mm
I would recommend the Medium height Leupold Ruger Rifle Scope Mount Rings, found by using the link below:
https://www.opticsplanet.com/leupold-ruger-rings.html?_iv_tube-diameter=30-mm&_iv_ring-height=medium
You would need Medium height for 44mm
Yes these rings will be good for the scope and the height is High for scope.
You need the 5k and 6k they are not a set
These are sold as a single ring.
Ruger factory rings are offered in a stainless, matte stainless, blued and matte blued finish. These are the stainless models and they will be the closest to what you are looking for.