It seem to me that the tubing frays less if you lightly hold the end together with the fingers of your left hand as you push the tubing on with your right hand. I push the tubing past the thread a little, make a couple of loose loops to hold it and then finish the rest of of the wrapping.

As far as the eye end, I start the thread over the top of the tubing where I want the head of the fly to be. Once I get the thread started and a few wraps on, I pull the tubing straight up and cut it off. The eye of the hook slips though the tubing so you can cut if very close to the thread. Then its just a matter of wrapping the thread to cover the cut. The tubing might twist a little as you start the thread, but you can straighten it out after you cut it.