For instance, can I take classical sheet music and assign the first violins to choral soprano, second violins to choral alto, violas to choral tenor, cellos to choral baritone, and double basses to choral bass, and transpose as needed?
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Only if the singers can sing the parts. My main argument would be the range. The range of the violin is way wider than that of the average soprano; thus, if the violin part goes on a really high note, a soprano might not be able to reach that one.
There are many things that can easily be played by a string instrument that can't be sung. So, some string section pieces might be able to be arranged for choral music, where others might not.
As Shevliaskovic notes in another answer, and indeed you acknowledge in your question by saying "transpose as needed," there will usually be some adaptation required to account for the fact that bowed strings and human voices have different capabilities. Some string music would be impossible to transcribe for voices without changing it considerably, since string instruments can, for example, play wide-ranging arpeggios very rapidly.
You may also find that instead of transposing, the music calls for revoicing. If both violin parts are well above the alto range, it might be better to assign them to divided sopranos rather than to transpose one of the parts down an octave. The cello can even go above the alto range, as it does in the aforementioned Barber Adagio, where the top note is G♭5 (g♭'', an augmented second below A=880). The violins are, at that point, on G♭6 and B♭6, well above soprano range, and the violas on D♭6, which is out of range for most but not all sopranos. I do not remember how Barber handled this in the Agnus Dei.
Also, don't forget that singers have to breathe. I've encountered more than one novice composition or arrangement that failed to take this into account.
Possibly. Look at each instrument's part. Could YOU sing it (maybe after a suitable transposition)? If so, your arranging strategy is worth further investigation.
Don't try to be TOO literal. For instance, strings can emphasise a note with a tremolo very easily. This doesen't have to be directly imitated in a vocal version.