[JohnKSa]

Does you employeer give any reasons as to why this rule exists? What if someone drops you off at work , take a bus or taxi, does the bus have to stay in the lot?

The reason given was that crossing the big street from the only nearby parking lot that's strictly off company property was dangerous for the employees. Drop offs are ok.

"Right to work" deals with required union membership.

Correct. When all the employees are required to be union members, there are usually rules imposed by the union affecting the "at will" nature of the employment. They're not strictly linked, but the one affects the other to some extent or another.

Even if I say Joe has a gun in his car, they will likely not forcefully search Joe's car.

That has happened at least once in the recent past at my company. Without getting into the details, a non-employee who wanted revenge, manipulated an employee in such a manner that the non-employee was certain that the employee would have a firearm or firearms in the car at a certain time, and at that time notified security that the employee had firearms in his private vehicle on the company lot. The car was searched and the employee was terminated. Security and the company management was aware of the manipulation aspect and felt it made no difference.

The search was not "forceful", there is no way they would search "forcefully". They ask permission to search and if you say no you are terminated for that offense. If you allow the search, it is not "forceful" but you will be terminated if any prohibited items are found.

No, they can decide based on a rumor to search any car they want, and only that car.

Correct. I've seen them do it.