The low E note in the first measures is a "drone" or "Pedal Point". It's named the latter because it was commonplace in Organ works where a low note would be held on the Pedals, while chords/melodies were played above it on the Manuals (Keyboards).
As a "non-chord tone", we call it a "Pedal Tone" most of the time (and it doesn't always have to be a low note).
In this particular case, since the piece is in E Minor, and E is therefore the Tonic, you might hear someone refer to this as a "Tonic Pedal".
It is, as I just mentioned, a "non-chord tone" which means, it's not part of the chord!
So for analysis purposes, we throw it out!!!
However, sometimes, it's possible for it to be a note in the chord and it depends on the context whether we throw it out or not.
Additionally, we don't usually label the inversions of the rest of the chord above the pedal point (another name for it) unless the lowest note of those upper chords have some interesting movement worth mentioning.
Measure 2 is a ii%7 (half-diminished 7) chord (F#-A-C-E) and the the first two beats of the measure have an "Appoggiatura" (also sometimes called an "Incomplete Neighbor"). So the G is actually a NCT as well. Thus we'd label the entire measure "ii%7" and mark the G note up top as the "NCT App." (but not concern ourselves with the inversion becuase of the pedal point).
Measure 3 is a V7 chord. B-D#-F#-A. See it? (throw out the low E). Oh yeah, the G note is an "upper neighbor" rather than a passing tone.
Measure 4 is also a B7 (V7) on beats 1 and 2, then it moves to Em (i) on beats 3 and 4
Now, measure 4 sets up an interesting question - as do the next measures - which have two chords per measure up until the cadence on G.
Though this isn't consistent throughout - the next passage has one chord per measure, then ones that change on beat 4, etc. it raises the possibility that measure 2 has two chords in it as well.
Now, it could be:
Beat 1 - C (VI)
Beat 2 - Am7 (iv7)
Beat 3 - F#%7 (ii%7) (no 3rd but we just heard it)
Beat 4 - F#%7 (ii%7) (complete this time).
However, in cases like this, for analysis purposes, it's often better to tell what the measure's "purpose" is.
This is textbook Functional Progression - i - iio - V - i. So your analysis is correct! But I'll continue in my long-winded, hopefully instructive analysis...
Now, in Tonal music, 7ths as functional chord tones, should resolve downward by step.
If the Am7 chord were Am, the G should resolve down by step, and it does.
If the F#%7 is a "true" 7th chord, it's E should resolve down by step.
Well, the low E doesn't, but it's a pedal tone, so we're ignoring that anyway (or are we? more on that to come).
But the E in the "tenor" voice on beat 3 - if that was a 7th of F#-A-C-E, then it would resolve down. Does it? Yes, ultimately, to the D# in measure 3 (also on beat 3). So while it's delayed, we could make a case for that resolution (especially since it was on beat 1 it would be "negated" by the pedal E).
But what about these first two beats? Does the low E Pedal count?
Well, often, if it's in the chord, we say yes. So if this were a C or Am7, or C to Am7, E would be a chord tone in each. I think it's safe to say it is in fact a chord tone in the first Em chord!!!
But you can see with the B7, it's totally not (and remember, this E should be sounding the entire measure).
If you look at measure 10, you'll see a similar idea with a D7 chord that changes to B7/D#.
Now, at this point in the piece the bass notes "count" because the pedal point really stopped around measure 5.
But look what the upper notes do here - it's that same idea of the G note as a Suspension that resolves to the F# chord tone.
VI6 - iv4/3 - ii%4/2 is a "logical" progression, it's a little too "minutae" to mean much.
iv is a pre-dominant function chord, and so is ii.
So really, measure 2 is "more about its "ii-ness" " than it is about these 3 different harmonies. For that reason, I see it as:
m. 1 - Tonic
m. 2 - Pre-Dominant
m. 3 - Dominant
m. 4- Dominant resolving to Tonic
Textbook tonal harmony - textbook functional progression.
m. 1 - i
m. 2 - ii%7 - with the Appoggiatura (inversion could be noted since it does resolve beat 3 to beat 3 of the next measure)
m. 3 - V7 (again, inversion could be noted)
m.4 - V7 to i
Makes the most sense to me. So I agree with you - it's "frilled up" with a bunch of non-chord tones, including the bass pedal point.