“Can a disc hernia cause leg pain?”

Absolutely if it impinges on a nerve

Yes. If the disc herniation is pinching a nerve it can cause leg pain. This is a very common problem treated by spine surgeons,

Yes

I believe you are talking about a "disc herniation" in the lumbar spine. A herniation of a disc in the lower back is when a portion of the disc exit its normal position and in most cases enters the spinal canal. If the disc material is large enough it can compress the nerves in the spinal canal and cause pain. The nerves in the lower back are the ones that convey information to and from the legs, so when they become compressed the nerve pain is present in the same pattern of the nerve tract down the leg. So, yes a disc herniation can produce leg pain.