No, no limit on the number of bags, but you may want to keep it somewhat manageable. Once you arrive at the port, you can drop your checked luggage off and turn them over to the porters and they will get them on the ship for you. THen you can just check-in with your carry-on bags.
What cruiseline are you going on? Some lines have self service laundry rooms with irons and ironing boards, others have a service where they will do it for you.
Your luggage should fit nicely under the beds, never had a problem there, if it doesn't, yes your cabin attandant can store them for you.