I think you'll find that a rack that fits a 29er is meant to clear a relatively large tire. So you're likely to have room. If you wanted to get a better idea you might be able to ask to measure the rack at your LBS or see if there are specs on Axiom's web site. Or you could ask on Amazon. The number you're interested in is the distance from the lowest dropout attachment point to the bottom of the rack over the tire. Then measure that same distance on your bike. Allow about a centimeter (1/2 an inch) for fenders if you think you'd like to use them.
Do the cables and housing look to be of a similar vintage to the bike? The rear brake usually has a longer length of housing and corrosion or junk (bug nests?) in the housing will drag on the brakes. As a quick check you could pull off the rear wheel and work the brakes several times – adding lubricant if you have some on hand – to see if they brakes start to free up. While you do this, watch the behavior of the inner cable, if it is hanging up on the housing you may see that it stays slack for a moment or takes up slowly when you release the brake lever.
If working the cable seems to help then the best thing to do would be to replace the cables. It wouldn't be a bad idea to replace the shift cables too. A little bit of lube on the pivot points wouldn't hurt either. Just keep it away from the rim and pads.