#1
What do things like d3, m5 mean?

They're intervals, i think, but what interval does each letter and number symbolize?
#2
They're used indicate a specific interval relative to a tonic (1, or C in this example).
These are the intervals typically used in Western scale and chord construction.. all with C as the root.

1 - Tonic = C.
b2(b9) - Minor second = Db.
2(9) - Major second = D.
#2(#9) Augmented second = D#.
b3 - Minor third = Eb.
3 - Major third = E.
4(11) - Perfect fourth = F.
#4(#11) - Augmented fourth = F#.
b5 - Diminished fifth = Gb.
5 - Perfect fifth = G.
#5 - Augmented fifth = G#.
b6(b13) - Minor sixth = Ab.
6(13) - Major sixth = A.
bb7 - Diminished seventh = Bbb.
b7 - Minor seventh = Bb.
7- Major seventh = B.
1... etc.

The m abbreviation means minor. m3 would mean minor third, m7 would mean minor seventh. (Note: Your example, m5 (minor fifth) does not exist.) d is used to represnt a diminished interval, d7 would mean diminished 7th, d5 - diminished 5th and so forth. a or + is typically used to denote an augmented interval, a5 (+5) is an augmented fifth. An upper-case M (M7 - major seventh) denotes a major interval.