The Memphis, Tennessee, school district is trying to steady itself in the wake of controversies that have rocked its operations side, potentially complicating its search for a new superintendent and renewing talk about putting the city's mayor in charge of the system. In the past few months, the district has been contending with official probes of how school construction contracts were won and how the food-services division was managed, and with mounting questions about lucrative, no-bid contracts awarded to busing companies. The questions appear to have reignited intermittent conversations about whether the mayor should have expanded authority over the schools, whether school board members should be appointed instead of elected, and whether the city school system should be merged with the county's.