Yes, a cat will help your rig run better and cleaner. Since these engines aren't race engines they aren't designed to work with an open exhaust, they work best with the exhaust components that came with them. They need some amounts of back pressure, a cat will add just enough as long as you don't open up the exhaust to much. The old cat should work fine providing you can still see some amount of light through it

You could increase the tube to 2.5" if you have free tube laying around, though 2.25" tubing IMO is better, inside diameter that is. I increased the exhaust on my '91 22re from stock to 2.5"I/D from the cat back and hated it, no low end torque and maintaining 55 on the highway while climbing hills was impossible. I dropped the tube down to 2.25" I/D and it seems to perform better. My tubing was also a straight shot, no bends at all, so the usual bends in the tube might make a difference with the larger tube.