Ahem, the progression C/A7/Dm/G7 does NOT follow the circle of fourths going from C to A, though the circle is followed from then on.

Note also that A(7) is not a native chord to C major; even in the temporary tonality of Dm, it's a harmonic alternate of the natural Am(7). While these progressions sound perfectly fine, what's happening harmonically in the move from C to C#dim/A7 isn't adequately explained. Yes, it works, but WHY?