Kenmore refrigerator makes loud clunking noise when motor shuts off, otherwise is functioning fine - is this sign of bad things coming on or something I can fix myself?

Something is loose like tubing or the clips holding compressor. This is serious because loose tubing vibrates and causes cracks. If the tubing is rubbing against something it will rub a hole in the lines. Probably what happened is that one of the straps holding the tubing broke from a kind of "dry rot" that happens with plastic exposed to heat over a long period. Unplug the unit and remove the front kick plate and rear cover. Clean it well with spray soap and water, then you will be able to see what you are working on and the compressor will run smoother. Dirt will also cause the compressor and fan to work harder and burn up. Keep it clean. Make sure to strap down anything that is loose with wire straps (plastic). Make sure to remove fan blade and clean it. Put a scratch on it so you know which side goes toward motor.