Using irons in cabins has been banned on my recent Celebrity (owned by Royal Caribbean) and P&O cruises so I expect it will be the same rule on Navigator of the Seas. P&O have self service launderettes with ironing tables and irons but Royal Caribbean and Celebrity do not and you will have to pay to have any pressing or laundry done.
No you cannot I'm afraid, its a fire risk an no longer allowed. RCI do not have a laundry room on Navigator you will have to pay for pressing.
There is an excellent valet service onboard and they will be able to iron any of your clothes for a very small cost.