??? Pretty technical questions unless one knows a helluva lot about the Shimano Nexus 3.
Why don't you just get an old three speed hub from an old three speed bike and be done with it?
Good luck finding a coaster brake 3-speed hub used, however. They are sought after by collectors.
Yes, you are correct, you cannot use a coaster brake with a chain tensioner. Look, if this is for a winter commuter, why not just get and old three-speed and use the frame? You can change the handlebars if you like.
I used a three-speed for a commuter for many years and it worked very well. I think you will like it too. In my opinion, the riding geometry on the old English 3-speeds was better for winter riding than today's mountain bikes or road bikes. They are more stable and easier to balance on ice and snow.