I don't think there's a right or wrong answer here. I listened to the live recording — the transitions to C major are effective and convincing, so I would say it's fine to leave it as is, at least from a structural perspective. (If you have meaning / symbology for each key signature here, by all means change it.)
Didn't notice. Although you played very rubato, and the tempo changes are not notated in the score, so it's possible the outcome wasn't what you wanted. From the casual listener's perspective, it sounded fine.
The best way I think would be to tie the notes outside of a tremolo, followed by the tremolo notation. Something like this:
Let's start with the musicality of the piece.
Harmonically and rhythmically, the piece was quite mature. You are not afraid of time signature changes (when necessary), and know how to get around from one key signature to the next. We stayed within the modalities (C minor, C major, etc.) long enough to establish familiarity — a very important concept in music. As mentioned earlier, the transitional passages were effective and organic, not forced or contrived. The music made sense.
There was, however, a notable lack of counterpoint or contrary motion; this produced a driving effect that never lessened throughout the life of this 10-minute piece. In other words, your harmonic movement was mostly parallel. (This was typically upward. The passages that had downward movement were usually not harmonized [e.g., mm. 44; 337 - 341], although there were a couple exceptions.) Of course, this doesn't mean the piece is poorly crafted! From this listener's perpsective, though, it became a bit too repetitive — despite the agressive nature of the movement. A couple brief pauses and a meno mosso section (or two) would have served this movement well, in my opinion.
Part-writing.
This is definitely an exciting piece to play! There are a lot of block chords here to really give the fingers a grip on the piano. It's fast and it's furious — a great combination to impress an audience!
I did note that you relegated accompaniment almost exclusively to the left hand. There are maybe two passages where the ol' mano sinistra gets to play the melody; otherwise, it's left making octave leaps up and down the bottom half of the keyboard. Now, certainly lots of well-written piano music looks like this (many Chopin waltzes and polonaises come to mind), so it would be wrong of me to criticize you for so doing. However, I would point out that melodies can sound just as cool with accompaniment above than below. Something to think about if you weren't already aware.
All things considered, this is an accomplished piece, and the issues (if they were even that) I took were mostly stylistic / preferential in nature. Fantastic job, and I look forward to the rest of this sonata!
Best,
Jörfi