Oh - and here's a question for those in the know: should the
back of the coat have (blue) turnbacks or not? It's
stupidly difficult to find pictures of the backs of uniforms

If I understand you correctly, yes. There is a slit at the back of the coat, the fabric from the bottom corner at the slit is brought forward to meet the bottom corner of the front at the side. The lining shows.
I found this image helpful:

Austrian uniforms differed in detail, but the basic tailoring is the same.