I am wondering what causes things to appear in slightly different positions on the radar than their actual location - and how to fix this.

Sounds like a misaligned minimap. This should help: Minimap Tutorial

So far this isn't working well. The texture file either comes out a solid color or comes out with the map but the map is scaled down and in the middle. I can't get it where the map is just big enough to fit perfectly in the texture frame.

Did you make sure to add a mapbounds region in Zero Editor? Otherwise any unit can have a scaled down-/ or up minimap with:

I thought about a mapbound region and I tried that but it didn't change anything. I am not sure how big it has to be and where it has to be. Also I do not think this issue could be solved by changing character ODF values.

The mapbounds region is a square where the minimap/radar texture will be projected. If you change the size, the radar will also change. The region must cover the entire map...actually the playable part only. Otherwise you would have a random radar projection without knowing its dimensions. It is recommend to have a mapbounds region, because this would make drawing the minimap much easier since you'd know where certain places might be at which corner/edge. If you have the radar texture already and want to scale/move some places you can simply re-scale, re-transform your mapbounds region.

ok, I got the mapbounds region to work (thanks, I had it as cylinder) but now I can't figure out how to adjust it correctly. Map items are either way off or shoved inside the center of the map.

BTW: This is for stock Bespin Platforms if that helps.
Edit: At this point things look better with regular map texture and no map region.