I think this is all well and good, but it's a terrible bore to read. Meant, of course, as constructive criticism:
For instance: I'm a life-long Star Trek fan, but even though I've seen every episode of every TV series, I haven't the slightest clue how the Enterprise flies through space. I know the ship is equipped with an impulse drive for sublight travel, and that the warp drive powers the FTL travel. The latter involves dilithium crystals and antimatter. In the franchise's 55 years or so, this is all anyone really cares to know about how the technology works.
The transporter beams are an even bigger mystery: they convert matter to energy and back again, and Heisenberg Compensators are involved somehow.
The functioning of a tricorder has never been explained. They just somehow provide lots of information, and tell you which direction the life signs are.
My point is that the technology is not the story; it's just the backdrop to the story. And to the extent I know anything at all about how the Enterprise operates, it's because that particular item played a role in the story at hand. A character didn't just stop in his tracks, face the camera, and give a 10-minute dissertation on how Starfleet recycles water aboard its flagship.
I'm not trying to discourage you from pursuing this story; I just think you're worrying about the wrong things. This passage tells me you've spent some time thinking about the environment of your story, which is great. But if the propulsion, autopilot, and water reclamation systems aren't vital to the story, then it's not information that I as the reader needs to know. Some people might disagree, but I genuinely regard infodumps as an assault against good storytelling; if you must do it, then this better be the most awesome infodump of all time. Otherwise you're simply committing the sin of telling instead of showing.
What interests me more is the story itself. You've hinted at it throughout this thread, and the premise sounds really intriguing. But it's the technical aspects of how the ship works on which you've been asking for feedback. I think it's great that you're keeping these details in sight. However, it's the human story that attracts me, not the technical specifications of the ship.
I encountered a similar issue in my "Eyes" story; I had a character on one planet who wanted desperately to get to another, and this at one point required me to describe her mode of transportation. So I made her cargo pod a character of sorts; she recalled how her father had acquired it at auction, how she had practiced for her pilot's license in it, and how one of employees had left his marks all over it. What I was trying to do was describe the ship not as the Omniscient Narrator, but has the main character saw it herself.
As far as your other ideas, I think the decision to write some hard sci-fi is awesome. Ultimately, though, the realities of interstellar travel are so daunting that you ought to allow yourself a few liberties. Maybe push the time farther into the future, mention a few technologies that may or may not ever exist — a time-honored tradition in even the best sci-fi.
Relying on technology currently available or theoretically possible, travel between stars is so difficult as to be prohibitive. Even an unmanned probe would have to be tiny and self-sufficient; once it reaches, say, Alpha Centauri the probe can't wait 8 years for an instruction from NASA on how to dodge an asteroid (4 years to transmit its signal back to Earth, 4 more years to receive the response). The probe would have to be equipped with an AI to deal with these scanrios on its own, and yet it would have to be tiny in order to reach high speeds with minimal energy. (Nevertheless, I've read if we lauch such a probe today, we might get useful data from it in about 40 years.)
For space travel, mass is an issue. Never mind the problems of escaping Earth's gravity. Even for travel within the solar system, mass is a major issue: the bigger your vessel, the more energy is required to move it — and stop it. To the best of my knowledge, NASA's interplanetary probes only have manuevering thrusters, used for making minor adjustments. There are no thrusters for propulsion. They build speed by making elliptical orbits around the Earth, building up more and more speed with each pass until they slingshot off into space. Deceleration is also an issue at the destination; a probe traveling at interplanetary speeds might overshoot Mars if it didn't use the planet's gravity to reduce its speed.
(If there are any JPL employees following this thread, please correct me!)
So, using your premise of a city-sized ship traveling between stars using technology that is already theoretically possible, the trip would be extraordinarily long even over a span of a few lightyears. And it would require an enormous expenditure of resources.
Someone else provided some math equations that supposedly prove you can travel 4 lightyears in slightly less time, thanks to relativity. I have no comment, except that I doubt these figures include time for acceleration and deceleration.
For instance, if your ship instantly jumped to near light speeds, the Newtonian laws of motion mean your biological passengers would be reduced to a gooey smear on the rear bulkhead. (The Enterprise is equipped with "inertial dampeners" to counter this problem.) Therefore you'd need to accelerate gradually… and that alone could take decades. Once you reach the other planetary system, you then need to decelerate gradually for the same reasons.
A smaller ship would be faster, and thus Interstellar had a crew of 4 people augmented by all those frozen embryos. Years ago I read a really cool story about an interplanetary ship whose voyage was about 300 years long; all of the passengers were embryos, and they were raised by robots once they reached the destination planet. (During that 300 years, humanity discovered FTL travel, and so the mission was obsolete before the embryos were ever born.)
So if you want to go with the city-ship idea, it might be better to gloss over some of the technical aspects of how this ship got where it did, and focus more on the human reaction of being far from your origins and having your entire hope for the future being predicated on a complete unknown. That story idea — heading to the unknown planet, with no way of turning back — is a great story premise, and it's this I want to know more about.