it doesn't necessarily take any energy at all to process information, but it does take roughly kT work of energy to erase a bit of information. It's related to
No, and yes, so long as you don't delete information.
Think of a marble-based computer, whose inner workings are frictionless and massless. The marbles roll freely without losing energy unless they are forced to stop somehow, but computation is nonetheless performed.
I don't know how to compute with marbles without mass and stopping. Marble computers I've seen rely on gravity and friction, though I'd love to see one that didn't.