So a mnemonic can be a 12 or 24 words coming in a specific configuration from BIP-0039 list of words.
Does that mean that a mnemonic could technically be the "key" to access funds in 0x0? ( https://etherscan.io/address/0x0000000000000000000000000000000000000000 )
A mnemonic is simply a way to encode a large number into a more "human readable" form. This helps prevent transcription errors when writing it down for backup purposes etc. Does that mean that a mnemonic could technically be the "key" to access funds in 0x0? ( https://etherscan.io/address/0x0000000000000000000000000000000000000000 )
The large number being encoded is generally used as a "seed" from which private keys are then generated in a deterministic fashion. With 12 words, we can encode a 128 bit (+4bit checksum) number... 24 words = 256bits (+8bit checksum).
I'm not terribly familiar with the underlying workings of ETH, so I'm not sure if there is a key that exists for 0x0 address... if there isn't a valid key, then no mnemonic would give you the seed that would generate it (as the key doesn't exist!).
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If say one person magically finds the mnemonic for 0x0, if they increase the wallet path will that give them access to 0x0000000000000000000000000000000000000001?
That's not how the derivation of addresses from a seed works. It doesn't create addresses in a logical "sequence order" like that... if you did actually have a mnemonic that generated 0x0, the next address is most likely going to be nowhere near that in terms of "numbering". In any case, 0x0 might not even be the first address derived from your mnemonic/seed... it could be the 10th... or the 100th... or the 143498543957036534th 
