I think that the transcription might be slightly off in places based on my recollection of this piece.
In measure 63, the Am chord I believe should actually be a C+ triad first (C E G#), and then an Am chord in the next measure. Basically, this spot ends with a deceptive cadence (an unconventional V-iii cadence) that is decorated with the V becoming a V+. The augmented dominant chord can lead nicely to the tonic chord because the raised fifth of that chord leads to the third of the tonic chord (almost like a leading tone effect). However, there is also a line cliché happening here, where the chord stays the same, but one note is changing by moving up a half step each time. The original C chord has the note G, which then moves to G#, and finally to the A in the A-minor triad. While iii-I might not be the most common chord progression to move into a new phrase, the iii chord shares two of the same notes as the tonic chord; the fifth of the vi chord is the leading tone, which helps lead back into the tonic.
Measure 92 is a little more confusing. The B natural in the melody line is what throws this off. We could have treated the chord as a iii* borrowed from mixolydian, but F-mixolydian doesn't have a B natural. Maybe we could analyze it this way, and the B natural could simply be viewed as a chromatic escape tone that prepares us for the B natural in the following V/V chord.
There is no tonal scale that has a B natural and an E flat (I'm thinking Eb is easier to analyze than the D# written since F major is a key that uses a flat). The order of flats added to a key are BEADGCF, so if we heard an Eb, we'd also expect a Bb.
This chord could be borrowed from the Lydian-Dominant scale, which is also known as the acoustic scale/overtone scale/Lydian b7 scale. If we base the notes around the scale degrees of the major scale, it would be represented as 1 2 3 #4 5 6 b7 (F G A B-natural C D Eb). In the chord (A C Eb), the C and Eb are tendency tones that want to resolve down by a half step to the B and D of the G-major chord. I think this is one of those instances where the composer went based more on the sound he wanted than straight-forward theory. Explaining this chord and melody with the Lydian-Dominant scale might be overcomplicating it, but it's another possibility.
Finally, I think that in measure 114, the A is simply a passing tone that leads the G-major chord from first inversion to root position while the other notes of the chord stay static.
Hope this helps!