Does the emf source live forever with a purely capacitive load?
Charging and discharging requires work and needs energy right ?


It depends. In the real world, losses always occur. So, yes.

In an ideal world, with a lossless capacitor, the energy you get back from discharging exactly balances what you put in to charge it. So you could run charging and discharging for ever with no loss of energy.


  If the emf source were ac mains, would I get any bill ?


It depends on what sort of customer you are. In the UK, domestic customers are only charged for real energy used, so if the meter was working properly, no, you would not get a bill. Industrial customers are charged for reactive VA as well, as an incentive for them to get their power factor reasonable.