Is broodiness hereditary? I'd really like to know!

Yes the instinct to brood and rear young is hereditary. Several of the common breeds, Mediterranean specifically have been bred and developed over the years not be to broody. So, Leghorns, Minorcas, Spanish, Anadlusians, Buttercups, etc. are all "non-sitters". American, Continental, Asiatic, English Classes are all "sitters". I believe that the gene for this characteristic is dominant so that crossing any non-sitter with a sitter will produce a sitter. If anyone has information to the contrary, it would be good to get the source.
There are exceptions to all rules. When I was young my father had a flock of non-sitter. Of 50 birds, 2 went broody. I have also had birds in sitter varieties and some birds just don't go broody. These rules are general at best but if your goal in raising your own birds is to have some raised by their parents you are best to stay away from the Mediterranean breeds or you could be disappointed. Of course, if all you want is lots of eggs, then go with the non-sitters, they are heavy producers.
Hope this helps.