They do tend to wear after awhile-my LSD 3.73 can't hold any torque from the turbo, really.
Your differential is a clutch plate differential, not a visous diff. That right there could be your issue, if you put viscous-type fluid in it. You can buy clutch plates for the diff as part of a rebuild kit, but I don't recommend doing it yourself unless you know how to set pinion lash. That said, rumor has it that you can also fit an extra plate in the factory stack to increase torque capacity.
the factory 25% is typically plenty, IF it is in good condition. Most are less than 20%. Some E30 M3's (along with e34 M5's, but that is a different story) came with 40% limited slip rears, but those are hard to find, and are not marked in any unique manner.
Z3 Torsen is great, and with the cover will bolt in, as Dave said. The nose of the diff is a little longer, so you will have to loosen the compression fitting on the driveshaft and slide it in slightly.