Answer 1

Nidoking and Nidoqueen cannot breed together, but Nidoking can breed with any Pokemon in the Field and Monster Egg Group. Nidoqueen cannot breed with any Pokemon for some odd reason no one can explain. So sadly, no they can't breed with each other.

Comments

And that is why Nidoqueen really sucks. No egg moves.

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Cough* cough* http://pokemondb.net/pokedex/nidoqueen/egg
When Nidoking breeds the resulting Nidoran can be male or female.

Answer 2

No, they can't.
Nidoking is in the Field and Monster egg group.
Nidoqueen is in the Undiscovered egg group.