Can a school district/municipal government attempt to negotiate a new contract with its employees to help control salary and benefit costs? 

A school district/municipal government can ask the union to negotiate a new contract but much like the inability to freeze salaries or benefits, no unilateral decisions are permitted under the Taylor Law (see question 1).  Neither party is obligated to negotiate until a contract expires.  This means that in order to negotiate a new contract prior to the expiration of an existing contract, both the local government and the union must agree to do so.  It is quite uncommon for a union to negotiate a new contract, especially one containing concessions, prior to the expiration date of the current agreement.