If your flight is inside the USA (e.g. Salt Lake to New York), will you be asked about your immigration status?
It is very unlikely. The TSA does not concern itself with immigration status.
At some airports, the Border Patrol had been known to question people at the TSA checkpoints, but this seems to happen only, for the moment at least, at airports near the Mexican border, such as McAllen and Brownsville, Texas.
There was also a single isolated incident of a flight from San Francisco being met by officers from the Department of Homeland Security and passengers being asked to show identification, though the legal authority for this was questionable at best.
In Salt Lake City and New York, the chance of being asked about your immigration status is exceedingly low.