Simply a minor scale
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The third scale degree is called the mediant.
"That would be A minor. Go a minor third below the tonic of the major scale to find the relative minor." Technically, there is no relative harmonic major to the key of C Major. The relative minor scale of C Major would the natural minor scale of A. A harmonic minor scale raises the 7th note of the scale a half step, giving us G#, which is not in the key of C Major.
Yes it is.The third is flattened (one half step lower) in a minor scale.
Because it has a minor third (eg in C minor scale, C-E flat is a minor third interval), and a minor 6th (again in c minor, C -A flat is a minor 6th). This creates the correct pattern for TSTTST1/2S
C is the third note of an A minor scale.