When all bitcoins are mined, why would people set their computer to do computational work required in the p2p network?
When all bitcoins are mined, then the big miners will probably move out of the market (if the transaction fees won't be large enough).
If big GPU/clusters miners move out, then difficulty will drop so much, that you willl be able to generate blocks easily using a single multicore CPU.
Are bitcoin owners required to pay transaction fees in the future to reward for this?
Not necessarily. If they won't be paying, and big miners exit market, then it will be very easy to generate blocks.
I think the market will self-balance automatically, and determine the best transaction fee.