Well, the other side of it on a carrier is that if your ship doesn't need to carry fuel for itself, it can carry that much more for its planes- in a long campaign, that can mean the difference between having to sail for home on a Monday vs. continuing to provide air cover the rest of the week.
Hmm, no, not even remotely close to reality. Ships are designed around space and weight considerations, not amount of fuel held, considering nuclear propulsion requires the same moving parts as a gas turbine, and you haven't even taken into consideration the reactor or cooling there, you've got big issues if you look at it that way! Nuclear propulsion requires substantially more space and weight than any current generation propulsion method, they had an advantage over old oil or coal fired steam ships though.
Ships can replenish fuel, along with food, at sea almost indefinitely, it's maintenance and crew fatigue/sanity that provide your biggest limiting factor. I've been at sea on an FFG that did nearly 80 days straight at sea, that gagged on it! So it's your logistical chain that is extended, but that's all you really get. There's really not much argument for nuclear except you reduce your immediate fuel burden, submarines are a different issue, but comparable in a lot of ways.