Is melanoma genetic?

Sometimes. There are a portion of melanomas that have a genetic inheritance, e.G, the dysplastic nevus syndrome. We are not yet certain about exactly how the inheritance works for this. Other rare inherited conditions also increase the risk of melanoma and other cancers too. Finally, there are a large number of melanomas that have no inherited predispostion as well.
Yes & No. All cancers occur because of damage to the dna causing mutations in important cellular genes. Genes that regulate the growth of the cells. Environmental carcinogens (sun, x-rays, chemicals etc..) can induce new cancers. Also there are inherited genetic conditions which can predispose individuals to cancer development. Patients with inherited susceptibility are at a higher risk from carcinogens.
Can be. Having a family history of melanoma does increase your chances of getting melanoma.
Yes. I'm not sure i understand the context of your question, but like all cancers, melanoma is the result of a set of genetic mutations and other abnormalities. We know about some of them, are still learning about others, and only recently can we begin to treat melanomas with specific genetic mutations.