Based on what I can dig up, the dominant fabric materials of the late Viking era through the Early Medieval period were woolen weaves and possibly linen. My questions is: what weight and weave would be worn on what part of the body? I am familiar with some hand knits and naalbinding for socks and hats and maybe shirts, but what patterns and weights would most central or western european kotes/tunics or hose be woven out of? Was linen just for undergarments?

I think you should ask an expert (?) on medieval clothing here in this forum.
Maybe he can answer your questions.

2/2 twill

I'd look here although its bit later. It doesn't change much

http://www.museumoflondonshop.co.uk/store/pro...Staniland/


then maybe the icelandic bog burials, although cant recall what linen is there. Skjoldehamn kirtle (1000-1210), Kragelund man's kirtle (1045-1155)

Lots of info here
I don't know enough to confirm the accuracy but it looks good
http://www.personal.utulsa.edu/~marc-carlson/cloth/bockhome.html

Gregory Liebau has previously shared this link on Armour Archive. If you want really detailed information, it's a good example of the sort of academic research into textiles which is available.
https://www.academia.edu/6039908/Textile_Remnants_in_the_Archaeological_Heritage_in_the_Charpathian_basin_from_the_10th-11th_century