An SX4 rear end will work as a direct bolt-in. If the ratio isn't the same, you can put it in 2WD long enough to prepare a replacement axle. Most mechanics would charge about 80 to 100 dollars to rebuild a rear end if you bring it to them already removed. At that cost I never bother doing them myself. Its an extremely messy job that can get complex when you replace carriers and need shims and such.
If the rear has substantial damage inside it then its honestly not worth repairing. The stock Eagle axle is an extremely weak axle. My immediate family has broken three of them because my brother and I are apt to jumping airborne and leaning the car over on two wheels. The thin axle housing flexs easily, breaking the axle and leaving your tire to just float away and fall off. If you drive like a stereotypical Grandma the axle will last forever, but where is the fun in that?
For the cost of the internal parts you can pick up an XJ axle, buy new perches and shock mounts, and then bring it to a mechanic to quick weld in place using your old Eagle axle as a template. The whole cost should be no more than $300. It can be potentially much less if you scour C-list for a cheap axle and know a friend who can weld. The end product will be light years ahead of your existing axle.
There are clutch style limited slips available, but the factory units from a Spirit or Concord are crazy hard to find and the new units are very expensive. Most people use the lunchbox lockers. They are cheap and easy to install and quite effective, but they do give the vehicle a "kick in the pants" feeling around curves. There are other threads here on the Nest about them if you search through.