Does this type of hemming have a name, and is it fixable by hand?
The "Stitching" on the edge of your sleeve, is actually a strip of fabric, that was folded around the raw edge of the loden cloth, and sewn to the loden cloth. 

This strip of fabric is called "Binding". (it's not piping, because there's no cord running through it. The puffiness comes from the thickness of the boiled wool loden cloth inside it). 

The binding on your coat has a knit construction, making it flexible.

If I wanted to fix this coat by hand, I would find some yarn (worsted wool, cotton, or synthetic) that matches the binding (yarn comes in a great range of colors and weights) and sew it over the binding with a close-set blanket stitch. The blanket stitch creates a binding, because it wraps around the edge of the jacket. So it will cover the frayed part of the old binding. 
https://s-media-cache-ak0.pinimg.com/736x/0a/d3/91/0ad391d598c823cf864f51ff6f30edfc.jpg

Blanket stitch is easy to learn - find a youtube video. Use a heavy crewel embroidery needle.

Fiber choice - For durability, avoid rayon. Nylon and polyester are strongest. Choose yarn or binding that is smooth/firm/tight rather than fuzzy/soft/loose, for dirt and abrasion resistance. (cuff is subject to dirt and abrasion). Avoid eyelash, chenille, and other novelty yarns. Worsted (tightly twisted) wool sheds dirt.

Binding examples


Jute foldover braid
http://www.dritz.com/quilting-sewing-supplies/home-decor/decorative-accents/trims/44220/
Foldover braid https://www.vintagetrims.com/collections/braid-flat  
Knit wool binding
https://www.etsy.com/listing/485483369/12-wool-ribbon-126-yard-knitted-ribbon?ref=market
Worsted wool twill tape http://www.wmboothdraper.com/TapeLace/indexwithnav.html?tapelace.htm
Bias tape http://www.joann.com/wrights-x-wide-double-fold-bias-tape/prd34710.html
One way to attach bias tape http://sewtospeak.blogspot.com/2010/04/how-to-attach-bias-tape-with-mitered.html


Yarn examples


https://www.craftsy.com/knitting/supplies/patons-classic-wool-worsted-yarn/46556
http://www.knitpicks.com/yarns/Brava_Worsted_Yarn__D5420219.html