So... I was wondering, what is a chord like C-E-G-C called? Is it a four note chord? I heard four note chords were sevenths (notes stacked in thirds). So how is this specific chord named?
I would think it is just a triad since really you only have 3 notes [C (twice), E, and G].
More specifically you have a C Major Triad. As long as you have those 3 notes in there it is a triad. I don't think it matters if you have a note repeated (at the same pitch if you are playing a guitar for example, or at a different octave as if you are on a piano for example.
The note that is lowest in pitch will tell you what inversion the chord is in. So, in your example, with C being the lowest note this is the Root position. Putting the E in the bass would make it a C Major, first inversion. G in the bass will make it a C Major 2nd inversion. sometimes the bass note will be written in with a "/" slash as in C/G (c major triad over G). In this notation the major and the triad are both implied.
It's just call a chord. Doubling notes does not change the name. In classical theory, there are some procedures to determine doubling if one doesn't what the music to sound as if a voice dropped out. In orchestral music, almost all doubles are fine as doubling a voice doesn't add a voice.
In playing accompaniments to a melody, one often doubles some note of the accompanying chords so that the 7th chords don't sound too heavy. (One may want them to sound a bit thicker to call attention to them too.)
It's a chord. More specifically a triad with a doubled root (in the case of C E G C). If the 'chord' contains notes only from a triad it would generally be considered a triad with doubled root/3rd/5th although is open to interpretation as with many chords.
If the 4th note is a note other than a triadic note, generally it would be a chord(add x); eg. Cadd4.