1
If I want to make a bitcoin transaction I need to wait 3-6 confirmations for my transaction to be accepted (by say an exchange etc)
Is it possible to run a miner so that it is listening out for transaction on my address, to have my transaction confirmed sooner?
Or at least have the first confirmation confirmed sooner?
(Plus would I be able to recoup some of the transaction fees?)
2
No.
The only way to speed up the first confirmation is for you to (not guaranteed, but generally works) pay a higher transaction fee or for you to own a significant amount of the hashing/mining power and implement rules so that your mining farm always includes your transactions. Keep in mind that even if you owned 100% of the hashing power (of which you always prioritised your own transactions), the best you could ever get is your transaction in the next block which will take 0 to 10 minutes.
Note that once your transaction is included in a block, it's generally 10 minutes per confirmation after that (unless the block chain forks and your transaction needs to be re-included in the new chain), so it would take from 50-60 minutes for 6 confirmations anyway.
At the moment, If I pay a transaction fee of 0.0003BTC (30 cents!) my transaction is normally included in the next block. A fee of 0.0001 can take hours.
3
Yes, there are a few ways you can do this.
First, you could solo mine using GBT from Bitcoin-Qt, and prioritise your own transactions. But unless you find a block, this does no good.
Second, and perhaps easier, you could use a pool like Eligius which offers prioritisation as a perk to its miners.
I don't know where you're getting 7.5 from, but that can't possibly right. The average time to find a block is ten minutes, so the average time to get a transaction in a block can never be less than that. – David Schwartz – 2013-11-28T04:32:40.027
@David: Well the difficulty is constantly lagging behind by a little, and the average time to find block is maybe closer to 7.5 than 10 min. – Stéphane Gimenez – 2013-11-28T11:05:43.250