Definitely don't weld that unless this will be a track only car. On a track, especially dirt ones - the tires can just be dragged around when turning. Do this on pavement and you could easily cause some headache for yourself.
They do make various LSD units that can be installed - some are mechanical in design (geared), some use friction plates. KAAZ, TRD, Cusco, Quaife LSDs are popular ones out there - run in price from $700-$1500.
Performance - all depends on what you want from the car and your budget. There are many "options" out there, from engine swaps to forced induction - all depends on how much you want to invest in the project.
Typically, the best bang for the buck on a Corolla are suspension and brake mods. Coupled with a good tire and wheel combination - the car ride and handling will be significantly influenced. Powertrain mods are next on the list and can vary significantly in price.
any differentials for the 1989 toyota corolla sr5? front wheel drive carbeurated four . the twin cam motor but carb inducted.90 horse stock.