@ehBeth,
ehBeth-
Thank you.
1) "Wouldn't you have to prove it was done for some malicious reason? or that they don't have another home in another state?"
I don't think so. Even if it was done for a non-malicious reason (e.g., saving money by non-paying in-state registration fees and insurance premia), I think the co-op could refuse to issue a parking space based on that the license plate was obtained illegally.
2) "Are they required to prove ownership of the vehicle they park in their spot?"
To the bast of my knowledge this is not presently done. However, I think parking spaces should be limited to those belonging to residents.
3) "Are the spots dedicated to units or are they available to anyone?"
Each parking space is assigned to the car of a particular resident.