Khachaturian loved playing with enharmonics - the violin concerto has runs where you get D# and Eb (or similar) in different parts of the same run - or worse, two different Bb’s, as the run implicitly moves through different keys as it goes. This is then made particularly fun in the lead-up to the cadenza, where the violin duels with the clarinet, and to sound correct, you have to explicitly coordinate on which key the various phrases are actually playing in (given it effectively switches implicit key faster than the explicit key signature). From memory, you end up with the clarinet deliberately playing very different enharmonics to the violin, giving it an incredibly otherworldly feeling.
edit: to clarify, you literally have to say: “so play this Bb as the Bb in a G-minor scale, and this Bb as the Bb in a Ab-major scale” or similar - as they have different frequencies. Or more accurately “play this subphrase as if it’s in G-minor, and this phrase as if it’s in Ab-major”. Despite the Clarinet having fixed stops, you still “lip” the notes up and down to get the right frequency.
No, but they can sidestep the need to be perfectly in tune within themselves by allowing the player to produce notes unbounded by discrete steps or subdivisions thereof, e.g. fretless string instruments and trombones.
Then you have the problem that the player will himself be off, unwillingly, most of the time. Often more than the offsets of 12-tet to the "ideal" note.
Maybe at first, but it takes all of two milliseconds to recognize "oh wow, I'm sharp/flat, I should move my hand a smidge further/closer". With more practice comes better muscle memory and better accuracy. Even high schoolers in marching bands can learn how to play a trombone in tune; a professional musician should have no trouble with that at all.
Any proof to this? 12 tet can vary by 15 cents from just intonation. Even an amateur musician can hear how out-of-tune a 15 cent difference is.
Judging from all kinds of out-of-tune players in live settings, and youtube videos (especially guitar, which I follow a lot) I kind of doubt that...
Guitarists may be out of tune, but chances are more likely you're hearing a poorly intonated guitar. You can tune the open strings perfectly, but if your string scale-length deviates from what your fretboard expects, you'll have notes that progressively get more out of tune the further down the neck you play. You can't correct this with tuning, you need to adjust the tensions in your bridge saddles, and most amateur guitarists are afraid to do this.
Also you mention live settings, it depends on how big the group is I guess, but at smaller venues and smaller bands the stage monitoring is often so bad you can't hear your own guitar.
This is arguably a non-issue for fretless guitars, but those take more skill to play (and I'd imagine getting some chords perfectly in tune would entail nigh-impossible hand contortions even without there being frets involved).
Generally the bend is done after the fretted note is struck as well. I guess it would be possible to always pre-bend to a given alternate note if you wanted a constant tone, but it definitely seems like working against the grain of the tool versus just using a fretless instrument.
In fretted instruments, bending is done for effect, not for adjusting each and every note.
Think it would be possible to mockup some really interesting tunings/temperaments in BitWigs grid or Max4Live.
In modern rock music, some musicians will change guitars every song to have sweeter chords depending on the particular chords in that song.
Rock musicians change guitars (in a live or even studio situation) mostly to get a different sound or a different tuning (like going from "standard" to an open tuning). Not for microtuning adjustments, or because they have moved the frets to match an individual piece.
Some prog musicians might do it, but it surely is not a "rock" custom.
In arabic music, on the other hand, or renaissance music, and other genres, it is, and instruments there often have movable frets.