For the potential energy I think it means the elastic potential energy - i.e. the work done in stretching a segment of the string by a length dL-dx against a tensile (restoring) force of T.
I'm puzzled why I got negged...to try and answer your question: vertical displacement doesn't appear in the expression because given a sinosiodal shape - the mass of the string below the equilibrium position is the same as the mass above it, so the overall change in GPE is 0.