Is the action of the city council constitutional?
Yes, because the statements demonstrate that the building inspector has an attitude toward a certain class of persons that interferes with the proper performance of the obligations of his job. 
Yes, because the building inspector is a government employee and a person holding such a position may not make public comments inconsistent with current governmental policy. 
No, because the statements were lawful comments on a matter of public concern. 
No, because the statements were published in a newspaper that is protected by the First and Fourteenth Amendments