Yes, we do that all of the time. If there is a restriction on doing so, it will likely be mentioned in the Navigator or they will tell you in advance.
I am highly allergic to all seafood because of the iodine. In fact, I do not even use iodized salt although my allergist said that it would probably be fine. I have never had a problem swimming in salt water although I very rarely do so as I do not like the residue on your skin afterwards. In any event, I have never had a problem with the water on board ship. I would imagine that the process of desalination removes most of the salt (hence the name). I personally find the water tasteless but for people who do not like the taste, it is likely due to the fact that it is different from what they are used to. I would never drink the water when I visited my parents because I thought that it tasted like swamp water and they thought that our water tasted like metal because of the high levels of minerals. Both waters were perfectly fine for drinking but we could each definitely taste the difference.
I find it fairly easy to avoid the seafood and I find that they are very good with allergies. I make sure to avoid foods in the buffet that might be alongside seafood as people may mix the use of serving utensils. In the MDRs for all meals including breakfast and lunch, as long as you have made sure to indicate your seafood allergy, they will cook your food in a different kitchen or kitchen area to prevent cross contamination. The only places that they cannot do that is the quick service places on deck.