The main problem in analyzing tonal music is that we mainly listen to relations between chords. For instance, in the following progression in C major, A major functions as a dominant of D (D is the dominant of G and G is the dominant of C):
C A D G C.
OTOH, in the following progression the same A is the subdominant of E: E A B E.
This means that if a song modulates or there's a tonicization [1] the same chord will have different tonal functions and we'll listen to it differently. Just counting a chord in a song may not be enough if they have different functions.The number of repetitions also matters. Tonally, the progressions C | C | C | G | G and C | G | C | G are the same as C | G. Is he eliminating repetitions in the analysis?
About using A major in C; you can use it as a dominant of D (see my 1st example) or as a chromatic mediant [2] in C major. Of course, in modern music you can use anything you want, but these two are the most common uses.
And, naturally, the types of chords used will vary according to the music style.
[1] http://en.wikipedia.org/wiki/Tonicization [2] http://en.wikipedia.org/wiki/Chromatic_mediant