I don’t know about the rest of your questions, but you can get a ticket for running a stop sign in a parking lot. It happened to my friend a coupla months ago. Needless to say, he was ever so pissed.
Well- you are in Canada. I simply don’t know, tehn. Here in CA, the answer is USALLY not. Some parking lots are so posted, but I don’t know how legal that is. Note I am talking about minor infractions only. IANAL, etc
A few years ago the Wisconsin State Supreme Court ruled that traffic laws could be enforced on private property if the property is open to the public, such as a shopping center parking lot. But they were refering to offenses like DWI. I don’t think a stop sign placed by a private individual is enforceable, especially the small, portable kind. I know I wouldn’t write on it.