Will a zero-fee transaction get processed faster if I send it straight to a miner I am running especially for that purpose?
This will let you include your own transaction in the next block you find yourself, without paying a fee (but the fee would be paid to yourself).  However, if you are a solo miner, it will probably be an extremely long time before you find a block yourself.  It won't have any effect on whether other miners will accept your transaction without a fee.

There's no free lunch as far as getting fast transactions without a fee.