Is it possible to recycle nuclear radiation/radioactivity (and/or the elements causing it?) and process it back into nuclear fuel?
Up to a point you can. The French IIRC, operate a major reprocessing plant. This does significantly extend the amount of uranium fuel available. But there's still a lot of waste remaining.
Cutlass - Burial at sea poses problems due to corrosion and the fact that what's buried way out there wouldn't stay buried if there was a leak. Even if it's thousands of miles from civilization, it could still affect civilization through ruined fish stocks and so forth. Plus, burying it deep under the sea bed is a major technical challenge.
I don't really see the problem as something that can't be lived with. The deep abyssal plains have very little life, and very little current. So anything there is just going to stay put for many 1000s of years.