We generalize several classical results about Schur functions to the family of cylindric Schur functions. First, we give a combinatorial proof of a Murnaghan--Nakayama formula for expanding cylindric Schur functions in the power-sum basis. We also explore some cases where this formula is cancellation-free. The second result is polynomiality of Kostka coefficients associated with stretched row-flagged skew Schur functions. This implies polynomiality of stretched cylindric Kostka coefficients. This generalizes a result by E. Rassart from 2004. Finally, we also show the saturation property for the row-flagged skew Kostka coefficients which also implies the saturation property for cylindric Schur functions.