Those smooth rib front tires are going to be absolutely awful in the snow. Not sure what all of your planned uses are for the tractor, but if snow plowing is amongst them, you'll want a more aggressive tread.
These will do well in all conditions. I have them on all of my tractors, and they look like a tractor tire: Vredestein V61
These will be great in snow, but IMO look a bit silly on a garden tractor:
Carlisle X-trac
For rear tires, you appear to currently have 23x8.50-12 tires, which are mounted on a 7" rim, so I would stick with that tire size. You really need an 8.5" wide rim to mount the 23x10.50-12 tires. I don't understand everyone's obsession with putting larger tires on everything. In the vast majority of situations, more flotation is not useful, and you will have to add more weight to get similar traction to a narrower tire. For this reason I am considering going down a size on my 782 when I get new rear tires for it someday (currently has 10.50 wide tires on it, and I have a set of rims for 8.50 tires).
R4 tires are good on a subcompact with front wheel assist, but I have heard many people say they are terrible on a garden tractor. I wouldn't use those either. For 23x8.50-12 I would lean towards
Carlisle Versa-Turf or
Goodyear R14 tires. I will likely pick one of those options for my 782. I have not used either personally, but I have heard great things about both of them.
I try to stay away from the Chinese brands- The all seem to run small and are generally poor quality rubber.