Is it genes or lifestyle or both that make it likely to get diverticular disease?

Both. There is a familial component, however the standard western diet is the most likely culprit - low in fibre, high in fat. If you have a family history, be extra vigilant about eating lots of high fiber products, drinking lots of water, and using supplements such as prune juice to help regulate bowel habits if needed. Avoid constipation, since this is the direct cause.
Probably both. Genetics and diet are probably the 2 main causes of developing diverticulosis.