Well, you have two issues here....the first being the 'hair like' stuff coming out of the exhaust. This is the lining of the inside of the exhaust that is coming out. I've seen this a few times and basically you can just pull it all out, or cut it off until it comes out more and repeat. It won't affect anything not being there, but some people have gotten their exhaust system replaced when this happens...it's all a personal choice.
The second problem, with the trans, is a bit more tricky. Since it seems to come and go, it really could be anything from a clutch, throwout bearing, shifter, or internal gear/synchro problem. It's very hard to tell without getting into it. This would be a problem you really need to have a trans guy take for a drive, or show the hard shifting problem to. It could be something simple and inexpensive, or it could be a very expensive trans repair.
GM Dealer Technician For 18+ Years
In the automotive industry for 20+