In classical theory, the necessity or lack thereof of a particular chord member is generally determined by the note's tendency to lead to another note. That tendency comes most often from the interval of an augmented fourth or diminished fifth. Enharmonically, those intervals are the same, but in context, they are not, and they resolve differently. In a "dominant seventh" sonority, the third and seventh of the chord form a diminished fifth if the third is below the seventh, or an augmented fourth if the third is above the seventh. In the diminished fifth, the upper note resolves down by step, and the lower note resolves up by step. In the augmented fourth, the upper note resolves up by step, and the lower note resolves down by step. What this means for the dominant seventh is that scale degree seven, the "leading tone," resolves up to tonic, and the chordal seventh, scale degree four, resolves down to scale degree three. Scale degrees one and three strongly imply the tonic chord of the key, so there is a feeling of resolution. In other seventh chords, the seventh still resolves down by step. It is possible in the dominant seventh to omit the root without changing the function of the chord. Often, then, what would have been the lowered ninth is added, providing another diminished fifth or augmented fourth (against the fifth of the chord). If I had a convenient way to draw all this out on a staff right now, I would.
In jazz, we don't always resolve chordal sevenths. However, we still hear the tendencies toward resolution as defining of chordal quality. Further, there are dissonances within chords that we do not necessarily want to sound, as you have pointed out. In those cases, we will keep the note that is more responsible for defining the chord quality (usually). So, in your examples:
- Eleventh: The third is omitted because of the dissonance with the eleventh. The eleventh is more defining of the chord quality because, without it, you would no longer have an eleventh chord. Further, the eleventh, being a chord member outside the triad of root-third-fifth, has a tendency to resolve down by step. It is resolving "against" the seventh of the chord (even though the interval between the two is a perfect fifth, not a diminished fifth). Because of that, the seventh also should stay. The remaining notes are less important, but to establish the chord quality, it is probably best if the root stays somewhere in the chord if possible.
- Thirteenth: Unlike in the eleventh chord, the eleventh no longer defines the chord, and it is a weak tendency tone, so it can (and usually should) be omitted in favor of the third. Typically, as in classical theory, when we can keep the third and seventh of a dominant-seventh sonority, we should. The thirteenth is defining, so it should be present. There are no other strong tendency tones, and you already have three notes of the chord (third, seventh, and thirteenth), so feel free to leave everything else out if it sounds fine in context. If you want to include those other tones, that's also fine.
- Minor Thirteenth: With the flattened third, the diminished fifth between third and seventh is no longer present. This feature completely eliminates the tendency of the third to go upward, and the seventh is now a much weaker tendency tone. On the other hand, the third is the only feature making the chord "minor." So, the third should still be present unless context makes it clear that the chord is minor. The seventh is probably more optional than in the Thirteenth chord, since its tendency is weaker and it does not define the chord when the third is present. The thirteenth must obviously remain, as it defines the chord. The eleventh goes away due to dissonance with the third, though the dissonance is weaker than before and can probably be tolerated. The root is good for defining the minor quality of the chord in some contexts. Other notes can be omitted.
- Nine Sharp Eleven: The eleventh is not a strong dissonance against the third here. The eleventh defines the chord, so it must stay. Since the third is not a problem and combines with the seventh to create a diminished fifth, both the third and seventh should stay if possible. I believe the fifth really should be omitted due to its dissonance with the eleventh. The other notes seem optional.
- Thirteen Flat Nine: The flat nine is now defining, so it needs to be present. The thirteenth is likewise defining and needs to be present. The root is a dissonance against the flat nine and is certainly (at least) optional. The eleventh is a dissonance against the third and should be omitted. The root and fifth are not completely necessary.
Note that your assertion of two notes being insufficient to represent a chord is inaccurate. Two-voice baroque counterpoint has clear harmonic progression throughout. The same rules apply when the chords are simpler than those you list. For example, in a major or minor triad, the fifth is not necessary. In a dominant seventh sonority, the third and seventh are necessary (as the tendency tones), while the root and fifth can be omitted.
Of course, if you are playing in an ensemble, other instruments may have chord members that you do not, then, need to double. Finally, if you are performing with a soloist, it is best to avoid using the chord tones that the soloist is using. It will be nearly impossible to do so in an improvised solo setting, but if you are accompanying a known melody line, this principle can guide your voicing choices.
I am not a guitarist, but I hope this perspective generally has been helpful. As on all instruments, technical convenience (or necessity) may guide your choices quite often as well. Don't feel the need to include too many notes, though; sometimes doing so just makes the texture "muddy." Let your ears be your guide.