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Considering that everybody has their own most comfortable singing range: when using the 'movable do'-system for solfège, which pitches does one choose to sing the various syllables?
Can 'do' be any pitch one wishes? i.e. does this mean everybody chooses their own key to sing in, transposing the music while singing? And, if so: isn't this confusing when singing with others and/or an instrument?
Or is 'do' always the C from the C-major scale, and does one choose the most comfortable octave?
Can 'do' be any pitch one wishes?, i.e. does this mean everybody chooses their own key to sing in, transposing the music while singing?
A movable 'do' can indeed be any pitch, however, in a group setting, the entire group must use the same any pitch. Otherwise, you get unintended multitonality, and chaos! If needed, the group's leader could pick a pitch as 'do', based on the range of the piece, and the ranges of the individuals in the group.
But usually pieces are already written in a specific key, which defines what 'do' is for that piece -- movable 'do' always refers to the tonic, or first scale degree (the key note).
This is opposed to the fixed 'do' system (as is used by some European countries), where 'do' always refers to the pitch 'C'.
Example: Consider singing the syllables 'do', 're', 'mi'. In a fixed-do system, this will always mean to sing C, D, E, regardless of what key you are in. In a movable-do system, it will always mean to sing the first three scale degrees of whatever key you are in. So in the key of C, it is identical to fixed-do (C, D, E). However, in the key of A major, you would be singing A, B, C#. In the key of Eb, you would be singing Eb, F, G. And so on...
Also, as you note, solfege syllables do not specify the octave. Thus, they are not enough to completely specify a melody by themselves. That is not their intent -- they are meant as an aid to highlight the interval relations within an octave.
Frequency vs Pitch Fixed vs. Movable 'do' says nothing about what frequency (in Hz) a given pitch (such as A) will be. To determine that, there are different pitch standards, the most widely-adopted (and the official standard for the US and Europe) being A=440Hz. For playing in early music ensembles, other standards will sometimes be adopted; e.g. A=415 is frequently used for Baroque music, and A=432 seems to be a popular alternative as well (especially in youtube videos, which apparently claim that that frequency is somehow more relaxing than others). But using something other than A=440 is still pretty rare.
Movable Do is called such because it is just that: movable.
Movable Do focuses on intervallic relationships. When you change keys, "Do" becomes the new tonal center for that key and all of the other solfege syllables are transposed accordingly. For example, in the key of D, "D" would be "Do". If we changed keys to "F", "F" would now be "Do."
Fixed Do focuses on pitch classes. In fixed do pitch classes retain their solfege regardless of context. For example, "C" is always "Do" regardless if it is the root of the chord / scale or not.
The fixed doh system as used in France, as one example, can quickly become more of a hindrance than a help. If a tune is in , say, Eb, then doh automatically isn't the root key note (tonic); that will be called Me flat. One of the helpful bits to a movable doh is the lack of sharps and flats - usually, in simpler tunes at least. Thus, the whole ensemble regards the key note as 'doh' and all the others follow it as if the tune was actually written in C.(C sort of being construed as a datum point doh). I suppose it's somewhat like an alto sax player playing a 'C' note from music, but actually, that 'doh' for him turns out to be an Eb doh in reality.From that point of view, in an orchestral score, there would be several different dohs to compensate for the different transposing instruments, i.e. in a piece in concert D, the trumpet doh would be E.
