No, they're not. They are, in the states I'm familiar with, part of the state statutes. They set forth requirements with which drivers must comply, as well as penalties for failure to comply.

The regulatory provision you cited to is a regulation pertaining to airline operators. Failure to comply could result in regulatory fines. Imposed on the airline. Not on the passenger. As a passenger, I can't be arrested for failing to comply with a regulation that pertains to airline operators.

As I said, and you acknowledged, an airline may elect to deny boarding to a passenger. But that's between the passenger and the airline - and whether or not the airline could deny boarding without breaching its contract with the passenger would depend on the terms of that contract.

All searches by government agents implicate the Fourth Amendment. Case law upholding a search in one context can not automatically be assumed to extend to a search in a different context. My understanding is that there is, as yet, no case law upholding searches at other than fixed checkpoints. Gate checks don't meet that standard.