Would an orchestral string player use an open E string to play any note in a major scale of F♭?
Is it possible to play a scale of F♭ major on a modern pianoforte accurately?
Which is highest in pitch; the middle E♭ in a C minor chord, or the middle D♯ of the penultimate chord of a perfect cadence in E major, played by the same player in a string quartet?
One of my composition teachers explained that string players will play in the direction of the notes. Heading toward higher pitches, they will play slightly sharper. Heading toward lower pitches, they will play slightly flatter. He made it sound like a function of the physical aspects of playing. The hand naturally pulls in the direction of which ever end of the fingerboard the player is heading enough to make just a little difference in the pitch.
He used this while giving advice about writing, especially in heavily chromatic or non-tonal music, where the notes do not necessarily have a strict function in the key. He said to write sharps when heading upwards in the music and flats when heading down, instead of using standard key signatures.
So, in short, in answer to the OP's last question, the Eb in C minor would probably be played slightly flatter than the D# in E major. For singers, I'm not sure, but I would guess that since sharps communicate raising a pitch, singers will subconsciously sing slightly sharper when seeing a sharp, especially if unaccompanied. I would wonder if the same is true for any instrument that can make slight changes to pitch using embouchure.
When it comes to pitch, the question of "what is the music doing?" is just as important as "what note is written?"
An orchestral string player probably wouldn't be playing open strings anywhere, unless directed to.
Whilst Fb and E are enharmonically the same, on 12edo properly tuned instruments, and here we'll assume the piano is tuned thus - which is a slight fallacy - an E note in one key will most likely not be played at exactly the same frequency/pitch as Fb in a different key. And you're hardly going to find a piece with both notes featuring in different places.
EDIT: comparing 12edo with just temperament (which often sounds more 'musical', Eb, as a minor third of C minor, is higher in JI, whereas D#, as major third of B major, is lower in JI than in 12edo. So, yes there will be a significant difference in the pitch, depending which key one is playing in. Obviously only for instruments which can (and do) play in JI. Which I think is the nub of your question.
As the other answers said already, string players generally avoid playing open strings at all, except for old music in historically informed performance. That's not a hard rule though. If a fast passage can be made easier by playing some of the less emphasized notes on open strings, then I don't think many players would have qualms doing so (those notes just need to be bowed a bit extra carefully not to stick out). In fast passages, especially played by a whole string section, pitch isn't so clearly discernible anyway. And for a passage in something as far out as F♭, the players would probably not have much motivation thinking too much about the mess of flat-signs and fine-intonation details. Most would probably just mentally translate “F♭ = E” at that point, and might, again, use the open string if that makes sense.
Things look differently for notes that are long enough for fine-intonation though. Orchestral players do intonate notes differently depending on context, even if they would be the same on piano. General statements as to whether a flattened note is higher or lower than the sharpened enharmonic are problematic though, because the factors needed to decide are also dependent on the environment. The crucial decision is whether you construct a note from Pythagorean or Ptolemaic tuning. I discussed it somewhat extensively in this answer.
To adress the concrete question
Which is highest in pitch; the middle E♭ in a C minor chord, or the middle D♯ of the penultimate chord of a perfect cadence in E major, played by the same player in a string quartet?
This depends on the interpretation. For both of these notes there are reasons to play them higher than the E♭ on a 12edo piano:
I suspect a first violinist or cellist might play the D♯ a bit higher that the E♭, a 2nd violinist or violist would rather do it the other way around. But more likely, it would just be different for every performance.
Fb is the same note as E. Although there may be reasons to notate one or the other from a music theory perspective, the notes sound identical. A pianist playing an Fb major scale plays the same notes as one playing an E major scale.
Likewise, Eb and D# are the same note and have the same pitch.
Question 1: Yes, with the giant caveat that string players rarely play open notes.
Question 2: The short answer to this question is yes. The long answer to this question is that it depends very much on what you mean by "accurately".
Question 3: By modern performance conventions they absolutely should be the same pitch. Perhaps in practice this will not always be the case because of the more distinct consequences of confusing a minor third with a major third, but as far as I know no definitive answer exists to the question "do string players play leading tones with more accurate intonation than they do thirds".
