1. what kind of triad is this?
a. major
b. minor
c. augmented
d. diminished

2. in what position is it?
a. root
b. first inversion
c. second inversion
d. third inversion

3. what is the root of the chord?
a. e
b. a
c. c
d. f

it seems to me that yes they solo on the chord changes.  coltrane for sure solos on the chord changes. instead of changing scales, they use a different method over the chord changes. this melody is very  unique but may often times even be considered simple, which is quite common in this modal  type. for example, there are changes from the key of minor to major in this piece, which makes it  unique in its own way.