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.