In other words, a C major chord is C E G, but when I see C written on the sheet music, I am free to play it in any rhythm I please, with any combination of C, E or G (CE together, then EG together, or C then E right after, then CEG after that, or perhaps as an arpeggio, etc.)? Total freedom?
No, not in all cases. First off, if you see
C for the chord symbol, then that means a C major chord in root position, which means the lowest sounding note has to be a C. If the lowest note were meant to be an E or G, it would have said
Second, just because chord symbols are used doesn't mean a rhythm is not specified. Two ways to do that are to refer to an earlier rhythm figure or to write rhythm slashes. Often, a rhythm is not specified, so yes it is up to the player to determine the rhythm, although "total" freedom is technically possible, but most rhythms will change the sound of the piece and might displease many listeners and/or other musicians in the ensemble.
What about playing only C in the bass without E or G appearing at all? Can I still say/claim that I am playing a C major chord just with that one note in the bass C? Or is it more accurate to say that I am not playing C major at all - I am just playing C?
That's not a chord. By itself, just a C does imply some kind of C tonality (usually - it can depend on the notes before and after the C), but all by itself it can't imply a major or minor flavor. If you add an E, then you have the major flavor and a much stronger implication of a C major chord, although it could be an A minor chord with the A missing or an F major 7th chord with the F and the A missing, etc. Those more remote possibilities are unlikely to be "heard" by listeners unless very specific notes come before and after the C and E.
If the chord symbol is
C, then the intention is for you to play a C, E, and G, with C as the lowest note. You can play around with the rhythm and you can arpeggiate it, but those are both things that generally have to fit with the piece, unless you're trying to challenge the original sound of the piece.