as for the kryptonite.. I would assume he would only be weak to that if bartron didn't allow him to adapt to it..
as a side note.. lex luther tried kryptonite on dd rex and it had relatively little to no effect at all on dd but it did on superman (obviously).....
so.. I'd have to assume that since doomsday is stronger than dd-rex in every way except mental capacity that he too would be immune to K's effects..
the solar radiation shouldn't effect him since his body stores energy internally... that's how he can drift without sunlight in space for so long.. he absorbs it like superman but his body is better at retaining it.