Intervals, chords, notation, and superscripts, take one

Now that I'm starting to play chords, I'm trying to understand some of the traditional nomenclature and concepts surrounding them. (If I'm going to play counterpoint eventually, I'm going to have to play harmonic chords quickly and easily first. I figure one of the steps along that path is to be able to see a chord on the staff, know what it is as a unit, and play it, without thinking about the individual notes. That means I've gotta understand the structure of the chords, at least the common ones.)

So I was trying to understand what makes the G⁷ and D⁷ chords what they are, versus the C and F chords with no superscript. After some Q&A with my dad, combined with some deduction and calculation of my own, I partly understand it.

My dad's first instinct was to say that G⁷ and D⁷ contain "a seventh that wants to resolve". This was way over my head, but after some pressing I got him to explain that the ones with no superscript are "Major" chords, and that they start on the note whose name they bear, C being the first note in C Major, F being the fourth. (They are typically played with the notes out of order, but fortunately I have enough math background to understand equivalence classes, so we didn't get hung up on that.) Dad also showed me the "fifth" major chord in C major, G, and revealed that there are no others, just those three.

Some counting reveals that the three major chords, starting from their named letter, go up four semitones and then three. This, apparently, is what gives them the "major chord" sound. (They do sound similar to one another, when compared to the other chords.) Further investigation reveals that forming a major chord in like manner starting on any note other than C, F, or G forces the use of a black key, i.e., in C Major it means an accidental. Hence, there are no other major chords that fall within the key of C Major, because any others would violate the key signature. One supposes that in a different key signature one could form major chords starting on different notes.

Similar counting reveals that the 7-superscript chords go up four semitones and then six, so they are similar to one another in exactly the same way that the major chords are similar to one another. I have not yet determined, however, how the 7 superscript denotes this. My dad said it means there's a seventh, but Wikipedia says a major seventh is eleven semitones, which is way more than six, so I'm not clear on the connection. Perhaps once I see some chords with other superscripts I will be able to deduce a pattern.