Hard to Say. Some families have predisposition for back ailments, but hard to say if disc herniation specifically is related to genetics.
Not really. Although genetics may make someone more prone to having a herniated disc (posture, ligamentous laxity, muscle strength, etc) the actual disc herniation is usually from injury or trauma.
Don't think so but. Chronic back pain is a very common problem for which herniated disc account a large proportion. We inherit some predisposition to osteoporosis and to osteoarthritis. If you find it common in your nearest family then you may have a predisposition. Herniated disks require some type of injury or trauma.