I'll try to answer this as best as I can, but it might not be as good as you're looking for, because this is a very tough question to answer! But I'll give it a shot. If it's confusing, unclear, or poorly organized in any way, just let me know, and I'll rework it.
Orbital issues
Okay, so you have Death Stars at the $L_4$ and $L_5$ points of Neptune's orbit. Excellent choices, because these points - unlike the other Lagrange points - are exceedingly stable. You won't have to do too much any stationkeeping or other maneuvers to keep these telescopes where you want them. Well, near where you want them.
A quick calculation shows that these telescopes are $2 \sqrt{3}$ times Neptune's semi-major axis away from each other - a mere $1.56 \times 10^{18}$ meters. That's 100 times as far as Earth is from the Sun - not accounting for Neptune's orbital eccentricity of about 0.008 (very tiny, fortunately for us, but something to take into account nonetheless). The eccentricity, combined with the distance, gives us a bit of a problem.
Why? Because there's no way in heck we can keep these things aligned. Put two small balls in Earth orbit, at exactly the same height, with an orbital eccentricity of 0 - but 120 degrees apart form each other. Statistically, there's going to be some wobble. Now multiply that distance by an incredible number, multiply the size of those balls by a huge factor, and consider that even if they stay where they're supposed to be, orbital eccentricity guarantees that the distance between them will chance over the course of one orbit.
The distance poses another issue: Timing. With distances this large between the "eyes", light will reach them at drastically different times. It takes 8 minutes for light to reach Earth from the Sun. It would take hours for a signal from one telescope to reach the other - and so measurements would have to be perfectly synched.
The fix for the other issues? Nonexistent. There is no way we can eliminate all these problematic factors. However, we can try to eliminate them. The best solution might be to attach tiny thrusters to all sides of each Death Star, in places that aren't being used for the primary mirrors. That will help a bit. Another possible solution would be to have each Death Star operate independently - essentially saying "Screw coordination" - and use computers to compensate, and automatically align images from the different telescopes. That could help, although the amount of data that would have to processed would be monumental.
Construction problems
I'm going to ignore for the moment the problem that building a megastructure like the Death Star is way beyond our technological capacity - both now and in the near future. We have to scrap the whole premise if we don't accept that humanity has outdone itself and figured out how to build something like this.
There is a problem, though, and that is the mirrors used in each telescope. Look at this image of the largest optical telescopes in existence (and no, telescopes using other wavelengths aren't necessarily any bigger, except for radio telescopes). See the yellow one?
I know, the image is tiny, and so the captions aren't readily visible, but the link should help. Anyway, the yellow one is the Gran Telescopio Canarias, and it's the biggest optical telescope in the world. At only 10.4 meters.
Yep, there are bigger things in the image. The Thirty-Meter Telescope, the European Extremely Large Telescope, and the Giant Magellan Telescope are all still planned . . . but not ready until the early 2020s. The large pale outline is that of the creatively named Overwhelmingly Large Telescope - cancelled. And the big white thing is the Arecibo Observatory, which - while, in my opinion, one of humanity's most extraordinary feats of engineering - can only detect radio waves.
Anyway, there's a rather large problem, which is that there is a limit to how big you can make mirrors. Notice how all of the big telescopes are actually composed of smaller mirrors. Whole mirrors that size simply can't support their own weight while still being scratch-free and in the desired shape (parabolic). Now, the biggest of these still-in-the-pipeline telescopes, the European Extremely Large Telescope - ironically planned for construction in Chile - is 39.3 meters across, and will be composed of 789 hexagonal segments. A telescope on your scale would require hundreds of thousands of times that amount - because the number doesn't increase linearly. There are advances in larger mirrors, but you'd still need loads of mirrors 20 meters across to create just one 160-meter-wide telescope.
Is this possible? Yes. Feasible? No. Operable? Most likely not. Each mirror segment needs actuators to help it adjust to make miniscule changes, correcting for various effects. The actuator system for such a large mirror would be enormous, and I doubt that you could successfully take many images without some problem cropping up with a segment or two - or maybe a hundred or two hundred. Complex systems are always susceptible to disaster.
Properties of the telescope
Neil Slater pointed out in a comment:
There are three basic issues which limit what you can see: Angular resolution, light collection area, and absorption/scattering of radiation between you and the target. A good answer needs to consider all three.
Perhaps this isn't as good an answer as Neil would like, but I'll try to get at all three.
Angular Resolution: The angular resolution $\theta$ of a telescope is
$$\theta=1.220 \frac{\lambda}{D}$$
where $\lambda$ is the wavelength of light and $D$ is the diameter of the aperture, which here we'll approximate as the same size as the mirror - 160,000 meters across. The wavelengths of the visible spectrum range from $3.90 \times 10^{-7}$ to $7.00 \times 10^{-7}$. I'll take a mean wavelength of $5.45 \times 10^{-7}$ meters, which gives us a mean angular resolution of
$$\theta = 1.220 \frac{5.45 \times 10^{-7}}{1.60 \times 10^5}=6.649 \times 10^{-12} \text{ radians}$$
That's really, really good. The smallest thing that can be distinguished has a size of the above formula multiplied by the distance to the object. We can't use the next formula given on Wikipedia, because that's for a microscope. However, we can see if it would detect a human on Earth.
Let's say that Earth and Neptune are at their closest distance - $4.35 \times 10^{14}$ meters. Put that in the formula, and we find that the smallest object one of these things could detect is
$$l=6.694 \times 10^{-12} \times 4.35 \times 10^{14}=2900 \text{ meters}$$
Oh, well. We tried. So that answers the questions of seeing living beings anywhere nearby.
Light collection area: I made a mistake earlier. Can you see what it is? Well, I'll point it out: I assumed that the telescope would be able to use an area of radius 80,000 meters to collect light. This may be a fallacy, although not a huge one.
There are three main types of optical telescopes (as I'm sure you know): reflecting, refracting and catadioptric. The latter uses both techniques; for our purpose, it is a bit too complicated. A reflecting telescope looks like this:
A refracting telescope looks like this:
The issue? Both are long and cylindrical - most times, they're larger than they are wide. If you have a cylinder with a diameter akin to that of the Death Star, it will end up being much bigger than the Death Star. So in reality, we're left with a much smaller telescope. But I think we may ignore that error.
MarchHo suggested putting a reflecting mirror at the telescope's focal point, as is often seen in radio telescopes. It's certainly feasible, and would mean that the telescope could be a lot smaller - essentially a parabolic mirror floating in space (come to think of it, the depression in the Death Star from where the laser exits is shaped a lot like that). It's certainly an option that would solve the issues presented above.
Anyway, we can use the figure of $D=160,000$ meters for our calculations. We might as well be optimistic. The collection area for a telescope depends largely on its primary mirror. We can compare two telescopes, saying that telescope $A$ gathers a certain amount of light compared to $B$, given by
$$\frac{\pi R_A^2}{\pi R_B^2}$$
Take the James Webb Space Telescope. Its mirror is 6.5 meters in diameter. This means that our telescope has a collecting area
$$\frac{\pi 80,000^2}{\pi 3.25^2}=6.05917 \times 10^8$$
times that of the James Webb Space Telescope. Does that mean its angular resolution is that many times smaller? No, because angular resolution scales linearly with diameter, whereas this comparison is a quadratic scale. But the difference is still extremely impressive.
- Absorption/Scattering of radiation: I'm not entirely sure what Neil meant by this. He could be referring to interstellar extinction, which can hurt a telescope's abilities. The formula for extinction due to (neutral) hydrogen is
$$\frac{N_H}{A(V)}=1.8 \times 10^{21} \text{ atoms cm}^{-2} \text{ magnitude}^{-1}$$
where $N_H$ is the column density of hydrogen and $A(V)$ is the extinction, in magnitudes. The telescope plays no role in this formula, but the distance it is looking at does, so when looking through thick clouds of interstellar dust, a telescope's view is severely hindered.
The effects of two eyes
Building one Death-Star-telescope is one thing. Building two - and operating them together, as a cohesive unit - is another. I alluded to some of the challenges before (perhaps it was a mistake for me to use the word "benefits" before), so now I'll got into some more detail.
Coordination: Earlier, I said that it would be nearly impossible to keep one of the telescopes stable. I stand by this, though you said that the "wobble" is predictable and can be compensated for. Perhaps you're right in this case, because even though we have to take into account all the nearby bodies in order to modify the movement of the telescopes, that's possible. If you can correctly work out the predictions, you should be able to align them.
That's not what I'm concerned with, though. I'm concerned about movements of less than one meter, and rotations of a tiny fraction of an arcsecond. Why? Because the size of the telescope - and the vast distances it is surveying - multiply each and every movement. Remember from above how the angular resolution is about $6.649 \times 10^{12}$ radians, and how over the space of 30 AU that can cover almost 3 kilometers? Well, if you want to look in a very specific spot, and you rotate one of them by a tiny angle like that, the image will be off.
That's okay, though, because you can compensate a little, and the target doesn't have to be dead center. But trying this with two telescopes is trickier. Sure, you can try to adjust for wobbles by computer. But to figure out how the telescopes are aligned relative to one another is very tricky, and would take many measurements. Possible? Yes. Certainly. But not easy.
Communication: It takes 8 minutes for light to travel 1 AU. These telescopes are 100 times as far apart, meaning it takes over one day for a message to travel from one to the other and back. So that means that communication between the two is not going to be as fast as you'd like. NASA engineers bemoan this delay when communicating with rovers on Mars, which - in comparison - isn't so far away. But 30 AU is one heck of a barrier to climb over, and it may pose severe technological challenges.
Maintenance: There have been 5 servicing missions to the Hubble Space Telescope to fix tiny errors that could have made some parts of it worthless. A trip up to Earth orbit - during the era of the Space Shuttle - wasn't too difficult, and repairs could be worked into a regular mission. This scenario is harder.
Care to travel 30 AU to fix a tiny broken mirror? I wouldn't, and neither would whoever's running this whole operation (I guess that's you!). A trip to Mars can take 9 months; a trip to Neptune would take much longer. So each telescope has to be completely self-sustainable. This means we effectively have to make each one a space station. I don't want to get into all those details; I'm sure you know the necessary obstacles there!
And by the way - yes, I considered a fully automated telescope. But with the communication issues, it would be best to have humans there at all times. After all, even one of the two telescopes would be an astronomer's dream - they don't have to be used in conjunction with each other.
I used the term "benefits" in my placeholder. So far, all I've done is list the challenges and problems, as I'm apt to do in my answers. So I'll end on a positive note - not a huge one, but one nonetheless.
Have you ever seen a 3D movie? It's really quite amazing, though I'm not a huge fan because I find them disorienting at times. But the technology in recent years is astounding, and has led to enormous advances in, among other things, virtual reality - which has been undergoing its own boom recently.
Many 3D movies work by showing two slightly off-set versions of similar images, creating an illusion of depth. Our depth perception stems partly from the fact that our eyes are offset - just like these telescopes! What does this mean? We might be better able to determine how far away something is, which would be incredible. Would this be very effective for faraway stars and galaxies? Not necessarily. Parallax may still be our friend for years to come. But for objects closer to the telescopes, we could more accurately measure their location. And with these telescopes spaced out so far apart, they could improve our measurements for parallax!
