Hi Kevin,
Lots of good questions in your video. I’ll address them one by one
Backside skin: If it’s only the edge/flap that’s coming off (which is common), just use elmers to glue it back down. Being that it’s dog/cat, no sense in replacing it.
Sao not fitting together correctly: I address the issue with a nagauta shamisen in this episode of the blogcast. http://www.youtube.com/watch?v=m4LPCdpBOv8&feature=player_detailpage#t=48s
Loose Tenjin: In most all cases, the tenjin is glued to the sao, it would appear that the glue weakened, causing the tenjin to seperate from the sao. In this case, wood glue should definitely be used to join them together again.
However, I have seen pictures of special shamisen where the tenjin appears meant to be held only with friction, like the mitsuori sao. Much more rare, and no doubt more expensive. Being that the sao/tenjin is made from karin, I’d highly doubt this is the case. Anywho, thought I’d mention it anyway.
So again, use wood glue.
Itomaki: If the itomaki is too big for the hole, slowly use sandpaper to until it starts to fit. (Go slowly. You can remove wood, but you can’t put it back easily)
I’d still say keep it. Fixing it is easier than it seems.