Will a zero-fee transaction get processed faster if I send it straight to a miner I am running especially for that purpose? Is it possible to get a transaction with no fee attached processed faster by sending it directly to my own bitcoind running with mintxfee=0?

Does the speed of my hardware play a role?

Is there anything else I need to set in bitcoind?

Is there another way of achieving the same results (fast zero-fee transactions)?

Thanks!
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.