No. You don't earn interest on futures contracts -- currency or otherwise. Likewise, you don't get the semi-annual coupon interest payments on bond futures either. However, ED or libor rates will be factored into the overall price of the currency future.
If a currency were to "sit still", the interest earned on a cash position should equal the profit earned from holding a futures position that's correctly priced.
Yes, a futures contract on a currency that offers higher rates than USD (NZD for example) will trade below the spot rate, and slowly converge by expiration. The difference = interest earned from holding spot. Reverse that for currencies with lower rates, like EUR.