Hello party people. I'm toying with the idea of making the setting for a TTRPG game a hoop/donut, but I'm having trouble thinking about how to map it. I want to 1) represent it in 2D, and 2) project it to the 3D shape. What I have so far for each:

1) So I'm thinking the best map projection will be essentially two trapezoids joined at the wide edge. That joint wide edge will be the larger equator (on the outside of the planet), and the top and bottom will represent the smaller equator (inside the donut hole). That way each latitude is the right relative length/circumference to the others. (Unless there's some reason the circumference would decrease non-linearly that I can't think of?)

2) This I'm totally ill-equipped for. There are tools that map equirectangular projections to globes for me, so I've never bothered to learn the actual math involved, or how to invoke such math on an image. I think wrapping up the above-described image into a cylinder (kind of a cone?) and then folded around into a donut will do it without much distortion. Am I wrong? How would I do it?

3) I might be back asking for guidance with weird coriolis forces and seasons on such a world. This is the best treatment I can find, but there some details I haven't figured out how to fill in.