In the first measure from this section of Schubert's Serenade, does the accidental on the ornamental C at the beginning affect the "regular" C that follows?
Yes it does affect it. The accidental always remains in effect for the remainder of that measure. However, if the two notes are of different octaves, the first accidental does not change the latter notes. If a G5 is sharped, for example, all remaining G5's of the measure will be sharped. G4's, G6's or G's of any other octave will be left natural. A natural sign has the same rules as any other accidental.