The 0x0 address in Ethereum has a lot of token stored in it that have been burnt in the past. However, it appears they burning is the same as sending to 0x0. In that case, could all of those burnt tokens be recovered if someone had access to the private key for 0x0?
Hypothetically speaking, yes. If someone managed to guess a private key to 0x0, then they would have full control over all ether/tokens belonging to that address. It is so unlikely that it's considered impossible because you would have to randomly generate private keys until you found one that generates the 0x0 address. Also, there might not even be a private key that generates the 0x0 address because of the non-1-to-1 nature of hashes.
Yes they would have access to all the tokens and ETH. The only way to carry out such attack would be creating keypairs until you reach that address'es keypair. But you must understand that achieving success with such attack is extremely unlikely.