Many good things happening here.
I think the tempo should be just a couple clicks slower (not much, just the tiniest bit). Regarding the opening, I think it seems a tad rushed, and a couple of the chords aren't played as prominently as I feel they should be - for example, the C major chord (the third chord of the piece) seems a bit glossed over compared to the previous chord. G7 leads to C major, so rushing through the C major chord doesn't make musical sense to me.
Also, once you get into the body of the piece, I wonder how much more lyrical you can make each individual voice. It kinda sounds like alternating notes back and forth. A slightly slower tempo will also help you take a little more care in mm. 21-24.
This work is highly polyphonic, so I recommend you try isolating various voices (such as playing just the right hand and focusing on how lyrical/independent you can make the two voices). Also, try singing different voices/lines. If you are self-conscious about your singing, just do it when nobody is around. That's what I did when I was younger! Of course, now I couldn't care less and I sing probably too much for my friends' comfort
Regarding the thirds, I don't think the fingering matters as much as how you tense up your elbows and forearms... I think you'll find much more success, and have less risk of an injury long-term, if you can do the rapid motion with your wrist as the fulcrum of the lever instead of your elbow. Of course, that's easier said than done... It would probably take a lot of practice and time to switch your technical approach, but since you are pursuing a music major, it's highly worth it.
People like to crap on Hanon, but maybe you should try out exercise 48. His advice right before the exercise is actually really good... But just remember the faster you go, the less up-and-down motion you'll want. Eventually, as you get to the second part of the exercise (which is quite similar to the Schumann passage in question), you'll want it to feel as horizontal as possible, instead of a bunch of up-and-down motions.
You actually achieve this (playing from the wrist instead of the elbow) quite nicely in the octaves of the development section. For the octave section, since the right hand is in good shape, I'd recommend focusing on your left hand more, and bringing out the counter melody and harmony changes more clearly... A tempo just a couple clicks slower will help with that.
This is a tough work, and there is still work to do as you have acknowledged, but you do have good chops and with good, smart (AND PLENTY OF SLOW) practice, I think you can work toward performing this very well.