Marion, I'll try to answer most of your questions. Once you have found the optimal spot to mount the contact pickup, you'll want to be able to return to that spot, without getting back into the trial-and-error process that allowed you to find it in the first place. Perhaps Graham's "grease spot" method would allow you to do this in performance, but I can't say for sure. It does occur to me that it would be awkward moving the pickup from one instrument to the other in mid-set, while trying to keep the audience's attention on you rather than on your equipment. It would also be noisy, unless you muted that channel (turned it off) before making the switch. You also might run into sound issues, since the way you EQ the pickup (gain, treble/mid/bass, etc.) for the fiddle might not work well for the guitar. I won't say that all this makes it impossible, but it sounds problematic to me.

I think it is unlikely that many people move contact pickups around to adjust the tone of their guitar for different songs. The best approach, in my opinion, is to find a location that gives you good balance across the tonal spectrum -- that gives the right tone characteristics and doesn't over-emphasize or under-emphasize particular notes -- then mount it securely in that spot and leave it alone. You will also want to make sure that the cord from the pickup is tucked away somehow so that it doesn't rub or bang against the instrument while you play -- remember, these things pick up ALL the vibrations from the body, without distinguishing between good and bad/planned and unplanned noises.

The PUTW is a contact pickup, not an under-saddle pickup. It is designed to mount on the soundboard (face) of the instrument, either inside the body or outside. Generally I think people mount them on or near the bridge plate, which is the wooden reinforcing plate on the inside of the instrument directly under the bridge. I have not tried a PUTW yet myself, so I don't speak from personal experience. But again, I hear good things about them.

If your desire to get by with only one pickup is based solely on financial considerations, I guess I'd recommend eating macaroni and cheese for a few weeks to save up the extra money to buy dedicated pickups (contact or otherwise) for both instruments. You may well be able to get by with the same preamp for both, with an A/B footswitch that allows you to switch from one to the other with a tap of the toe (yeah, the A/B box will cost you a few dollars more -- maybe $25.00 US). I hate to counsel you to spend more if money's tight, but I think you'll be happier in the long run. Good luck.

[As an aside, I didn't exactly follow the reference to velcro in one of the postings above, but my guess is that the velcro strap mentioned is intended to go over the top of the pickup to hold it down, not between the pickup and the instrument. Putting it between them wouldn't work at all, because, as Graham mentions, you would not have the close contact necessary to transfer vibrations from the instrument directly to the pickup.]