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Well, I'm assuming by convention that the 7th is flattened, as the notation "C7" does.

I think the convention for chord notation is different to the convention for scale degrees. The 7th note in C major is B, but a C7 has a Bb.

TBH I think you should delete this post. I think I would have been confused had I read it when I was learning harmony theory. Please see http://www.idlewords.com/2005/04/dabblers_and_blowhards.htm.

Major 7 chords are explicitly written as such. So in the case of a C chord with a major 7, it would be noted as Cmaj7.