You will probably need it to pass emissions but you don't need it for the car to function properly.
If you have a non-turbo I wouldn't recommend going without a muffler because the exhaust would be way louder than you want it to be. If you have a turbo the turbo actually helps muffle the engine noise and you could run pipe straight back. 88hybrid runs pipe straight back with his turbo.
There isn't much of a performance gain to speak of by removing the catalytic converter (I don't believe) so I would not take it out of play just for performance but it is up to you!
I put a high flow cat on my n/a because my old cat was rattling a little. I got a magnaflow and it sounds nice and the car runs smooth.
It's a direct-fit cat, paid $118 with the shipping. A universal high-flow cat would've been about half the price but I didn't want to have to weld or cut anything so that's why I paid the extra cash for the direct-fit one.
I don't think it increased performance so much as helped regained some from the old shotty one.
I don't have any dyno graphs, but it just makes sense. Kinda like replacing'removing your muffler....remember, your cat is the same interior diameter as the muffler that you replaced that added hp. Mybe it's just me, but it makes perfect sense, and I know my old Mazda felt a he!! of a lot more powerful when I removed the cat from it.
the result from this on a n/a z31 would only be noise. there would be no noticeable gain from removing the cat unless the car was heavily modded. otherwise it wont make a noticable gain. i had the cat taken off when i got my ex. and i couldnt tell a difference from the stock(perfromance wise)