Generally, the root will be in there SOMEWHERE, but maybe just not in the lower register of your voicing...
The 3rd will determine if the chord is major or minor, but, yes, can be omitted (if playing root/5th power chords, or octaves, for instance). If a part needs to be DISTINCTIVELY major or minor, you'll need that 3rd somewhere.
The 5th... hmm... I'm sure there are tons of voicings that do not use the 5th (again, octaves), but I'm not sure that you'll keep enough structure without it--what am I trying to say... I guess it would be easy to confuse a 5th-less voicing as another... ahh, I don't know what I'm saying. I just picked up my acoustic, and after a few little plucks... yeah, throw that 5th out... lol
As for the second part of your post, I tend to just use smaller pieces of larger chords when looking for sparse sounds.
Such as, in an E-shaped barre, just using the fingering formed on the the 2nd, 3rd, and 4th strings of the "full" chord etc. If that even makes sense.
For instance, you need an Am7? hit this:
----
-8--
-5--
-7--
----
----
That's just the root, 3rd, and the 7 (actually, the b7).
Anyway, that's how I roll. I'm not wicked-technical, honestly.
I'm sure the theory gurus will be in here shortly lol
