spot yield curve

1)
Current yield curve are basically the forward rates, right?
so if we are given a current yield curve, we can simply use those values(instead of calculating forward rates) to calculate the variable rate payments that bank will make to us each year in return for the fixed payments that we make to the bank each year?

2)
when calculating the fixed interest rate we assume that

pv of fixed interest payments= pv of variable rate payments.

in calculating the present value, we discount using the spot yield curve (not the current yield curve).Why is that so?