Is there any chance, although extremely low, to generate the private key for the 0x0000000000000000000000000000000000000000 wallet? Does the system allow it at all?
This type of attractive address names called
vanity addresses and you can use https://github.com/MyEtherWallet/VanityEth to generate them. If you are very very very lucky one day you'll get it.
Yes it's possible in theory, but the probability of generating the private key for the 0x0 address (and any others) is:
Near enough to zero you could convert the entire solar system to a computer and try until the end of the universe and never succeed.
Yes, the protocol allows it: the protocol does not have any special treatment that prevents the zero address from operating like all other addresses if its private key was found.