Do tires with worn sidewalls need to be replaced? My bike needs a bit of a tune up, particularly the tires since the back tube has a hole in it.  The rims on the bike are 26"x1.5" and the current tires are 26"x2" and both have great tread, but the sidewalls look like they're worn through in spots (I can actually see through the fibers in a few spots of the sidewalls).  Should I replace the tires, too, or is replacing the tube sufficient?  Also, is a 2" tire OK on a 1.5" rim?

Here's what it looks like:
Yep, that tire needs to be replaced.  (It appears that either it's been run underinflated or you've been using your foot on the side of the tire to brake a lot.)