You are right again, @howard - I should have formulated my statements more carefully.
Once you know all the notes of the C major scale, then you are “home-free” in any song in C major, where the chord progression uses chords from the diatonic family. So, I should have made the clear distinction between the chord F major (7) and the scale of F major. While you can use all notes of the C major scale over an F major chord (then you’d play F lydian), not all notes of the F major scale are part of the C major scale.
So, yes, the notes ARE valid for an E min chord and an F maj chord, but NOT for the entire E min scale or the entire F maj scale 
While somewhat confusing, this gives you choices over what to play over a given chord. So, if we have an E min chord (and the song is in the key of C), then you could play the E phrygian or the E minor scale over that chord. If you encounter an E min chord, but the song is in the key of D, then you could play the E dorian or, again, the E minor scale. And so on…