I'm guessing translated literally this means 'the man wants that he does cook the egg', clunky as that sounds? The phrase να μαγειρεύει is subjunctive, yes?
Well, this sentence per se is not something you would get to hear that often. If you want to emphasize the meaning (in your sentence, for example), you could say "Ο άντρας όντως/πράγματι(=indeed) θέλει να μαγειρεύει το αυγό. ;)
It is! The nominative and accusative for neuter gender nouns (nouns that use αυτό/το) are the same. Το αυγό/το αυγό, το παιδί το/παιδί, etc.—compare ο άντρας/τον άντρα, ο σκύλος/το σκύλο.