Yes, but did he violate the law? Those who argue that the government is not prosecuting him because he is a member of the protected class (apart from arguing that he should be prosecuted for political purposes) disregard the fact that any self-respecting defense attorney could drive a bus through the government's case.
I really don't know. Lying is not necessarily against the law but there is that pesky background check that he signed (and living in CA, I have signed several of these forms). Is that considered a contract of sorts?
It seemed to be enough of a question that the gun shop owner didn't want to risk bringing trouble on himself/his business.
This might be an excellent debate for law students.