Tash, remember this basic rule of thumb: if there is an NCT (non-chord tone) that creates a seventh, if it is a dominant (V) chord, then it acts as a seventh. Otherwise, it is just an NCT.
Now let's analzye some things - I've put up my analysis to the best of my abilities here:
https://tinypic.com/4q0owpThe unorthodox progression in the first measure (which I will call the measure after the pickup measure, which doesn't seem to be the way they labelled it) leads me to believe that the V > iii is possibly just a passing figure, although there is no stepwise motion and no suspensions... oher than that I don't know why Martin would have done this.
in Bar 3, the third chord could be interpreted two different ways, depending on how you view the escaping tone (D). If you think the D is an NCT, the chord is a viio in first inversion, otherwise it is a dominant second inversion with a seventh. Although either progression works, the first seems more plausible because doubling the second scale degree in the dominant chord is not as typical as the second in a seventh, so I would view it as a seventh with an NCT.
In bar4, I'm not sure why you had labelled the second chord as a I, but it is actually a dominant with a, 8-7 suspension, which resolves (C down to B) in the next measure.
In measure 5 (labelled 6) you named the second chord a vii yet vi does not lead to vii - the eigth notes are actually passing NCT's (which I put in parentheses) put on the downbeat, which actually creates a ii7 on the '&' of beat two.
Be careful in measure 6 where the F becomes naturalized on beat 4, creating a dominant chord as tonicization to C major (V/IV) which indeed leads to a IV in the next measure.
If anyone notices that I did something wrong or disagrees, let me know. Does this resolve some issues Tash?