When you set up Bitcoin mining, is it possible to mine only a specific transaction or is it random? Thanks!
It is but it is pretty impossible for you to mine a single block.
If you have mined that block though, you can include single transactions, even if the fees are below industry average. That is why ViaBTC has this free transaction accelerator, they simply put your transaction into their block once they mine one. BTC.com also has a transaction accelerator, but that site charges you for doing it whilst Viabtc doesn't.
But again, are you really going to mine a block? unlikely.