The basis is the spread between the two floating rates.

The option is always worth as much as it's payoff as at that instance you could exercise it and get the payoff. As such you need to use this payoff at that node for your calculations.

It's the dollar value of a basis point move.

As far as I knowFuturesC + Xe^(-rt) = P + Fe^(-rt)ie strike and future price discounted at risk freeFXC + Xe^(-rd*t) = P + Se^(-rf*t)ie strike discounted at domestic rate and FX discounted at foreign rate