The sax can be a demon to play properly, it will warm up over the course of playing so you will end up going flatter and flatter as you play longer but you can correct for that with your embouchure or alternate fingerings.

If you can re-tune (fourth line 'F') after you've warmed up the instrument. You do that on that F because it is roughly in the middle of the range of the instrument meaning you'll be off the minimum on the other notes. If you tuned it on the highest or lowest note the other end would be off considerably.

The sax is a wonderful instrument but it is definitely a finicky one, I've never found one that had all the notes in tune when played without corrective measures, and I'm not even sure that it could be done, the flexibility you get in making it go sharp or flat on purpose comes with the price of having to correct almost every note you play.

You'll be unlikely to be more than a quarter-tone out of tune though, no? I used to practice saxophone with a tuner to enforce intonation consistency; I could intentionally bend notes further, but I don't recall the tuner ever picking me up on the wrong note when I was playing naturally.

I coded for transposing up or down by an octave, but not by other intervals. I imagine it wouldn't be hard for somebody to fork/modify the code to transpose by thirds, etc.