In 18th century harmony, which is the basis for most modern and pop music today, there are certain tones of a chord which are considered more important than others.
The only difference between a major and minor triad is the 3rd. In the case of a C major triad (C E G) the E is the third. A C minor triad is C Eb G. The essential components of either is the relationship between C and E or E flat. The 5th (G) is relatively non-essential. So harmony often gets by without the 5th of a triad.
Similarly, for a 4 note chord, called a 7th chord, the 5th is the least important tone and is the first to be omitted if you are short of voices.
For a dominant 7th (C E G Bb), if you are shorter still, the root is dropped and the tritone interval between the 3rd and 7th can convey enough of the flavor.
Power chords, which don’t figure much in 18th C harmony, omit the 3rd entirely, which means they cannot be called major or minor. Their ambiguity is absolute.
The blues make the 3rd into a pitch that can “waffle” or “bend” between the two intervals, or sometimes include both pitches (typically in different octaves and/or different instruments) at once. The ambiguous 3rd is a primary aspect of the blues, although not the only one.
So, to answer the OP:
1) What intervals are most common?
3rds, or their inversions, 6ths. A minor 3rd inverts to a major 6th, and vice-versa.
2) Do the choices of intervals help to establish a “sound” like Bluegrass or Gospel or Rock or Old Time or other genres?
To a limited degree; see the note on blues above. Some styles typically have a harmony line ABOVE the melody, but most is below.
3) How many different categories of Two-part Harmony are there?
I’m not sure what you mean by this other than what I already wrote.