paradox

is paradox hereditary, or something that just pops up every once in a while?

It's considered non-hereditary (hence, paradox-something that doesn't make sense but still exists) so, unfortunately, it can't be reproduced the same way recessives and other genes can.

(although there's some speculation that it may be multiple complex genetic combinations that cause paradoxes rather than simple genetics)

It is also believed to simply be a chimera.