Hard unpasturised like parmesan is fine. Italian babies really are weaned on it from an early age (6 months) as was my ds. Now he dislikes any melted cheese etc but loves parmesan in pasta!
There may be a case for not giving your child a taste for it too soon as it is pretty expensive .
Tuna is OK though I believe the possible mercury levels mean it is not recommended that young babies eat it too often.