Yeah, we discussed a lot of these codex entries back in the old Ark theory threads. Honestly, I hope Bioware read those discussions, because people really did come up with some creative ideas around them.
I think all of the problems have pretty much been solved now. Whether or not Bioware utilizes something as clever is another story, as I'm almost positive they will just hand wave it away.
1) Drive discharge - the codex says the Citadel and other large "deep space stations" somehow utilize drive discharge facilities that don't require a planets magnetic field or direct grounding. The Arks could use something similar. The Reapers also seemingly don't discharge, and although the timeline makes it nigh impossible that they are using Reaper tech to do it, it is just another example about how the lore doesn't forbid FTL without discharge in a planet.
2) Fuel - this first seemed like an insurmountable problem, but it actually isn't. See, here is how FTL in mass effect actually works: Electricity is run through element zero, which creates a mass effect envelope and lowers the mass of the ship. Then, a fuel source (antimatter, fusion, etc) is used with conventional engines to accelerate the ship to FTL speeds (faster than light compared to the speed of light in a vacuum, still slower than light in the mass effect envelope). The ship accelerates halfway, then turns around and decelerates the other half. Why? Because mathematically this is actually the most efficient and fastest way to travel in space, and it is exactly how we would do it in real life with sublight travel too, if we had engines that could actually accelerate that long.
But here's the thing, you don't actually need to accelerate that much. That's just how you speed the voyage up. All you need is a source of energy to apply a continual current to the eezo and an initial acceleration at the start of the journey and a deceleration at the end, with a constant velocity in between. Thanks to Isaac Newton, you'd keep going at that velocity until you decelerated or until the mass effect field collapses which catastrophically snaps the ship back into a normal velocity, per the codex, killing everyone on board.
This is basically "FTL coasting", it is 100% allowed by the lore and it would allow a voyage to Andromeda still in a comparatively short time, though probably over a millennia compared to 570 years. The actual time would depend on the final velocity reached. The best part of this is that you only need "fuel" to supply the current and initial acceleration and final deceleration. This is still substantial, but not insurmountable compared to a trip where you use fuel to continually accelerate halfway and decelerate halfway.
3) Construction - this could be addressed in any number of ways. Personally, I think the most brilliant idea one of us came up with was what Han Shot First came up with:
The Reaper Wars was not the first time the galaxy faced extinction. The Rachni Wars were nearly equally as apocalyptic. For reasons explained in the codex/lore and unnecessary to rehash, it was basically the same scale and magnitude of the Reaper War, but drawn out over centuries instead of months. Still, the end result was that the Citadel species were facing extinction. If the Asari and Salarians built these ships during the Rachni War to flee the galaxy if the Krogan failed, and then mothballed them at the end, it would absolutely provide an out for the construction timeframe and the lack of believability.
Most everything here I can see except the Rachni war bit. It would, in my mind, make more sense that the Ark ships weren't of any species' current design, but remnants from another cycle. Whether that would mean the ships themselves, or remnants of ships that were lost/destroyed through whatever means, it wouldn't matter much, though I'd find it very questionable if they found fully working ships.
The size of the things is questionable, but with the remnants of the Reaper and the weapon that destroyed, I'm on board witht he idea that some places of the galaxy, where you don't typically travel to or through could house long lost secrets. Essentially it would come to the idea that one or several cycles decided that they couldn't beat the Reapers, and tried to escape, but ultimately this endeavor failed/was lost.
Repairing such things, or building them from plans, still seems a vast and unlikely task, however. I couldn't see it involving any normal military or government body without it being known by an entity like the Shadow Broker. The N7 stuff showing up seems to suggest this line of thinking just isn't likely, because it would seem more likely this organization was operating in areas of the galaxy that people don't go to or through, speaking of how they tried to chart FTL paths, for example, there are places you just don't go. So finding something like this in such areas, and building up such areas, and just not leaving them?
Of course this would mean they were repairing or building such tech with no intention of telling anyone else, and it may be questionable whether they were doing it because of the Reapers, or not, depending on the information they had.
It all seems very unlikely though, but it's the sort of stuff I'd come up with if I were trying to force the concept into something it probably didn't fit.
Still, if I were to go the hand wave route I'd abandon Mass Effect technology entirely, and go for something more exotic, while making every member on the Ark ships part of a isolationist shadow organization. The technical and plot details would still be very unlikely, however, and I'm just not sure any of it could make sense no matter how hard I tried. Hence why I couldn't justify it as anything more than hand waving with, basically, "Space Magic."
I suppose you could go another way and say the ships weren't made in our Galaxy at all, and that the people on them simply hopped on for the return trip. You could pull a bit of a Rendezvous with Rama sort of deal where the people on the ship are samples along for the ride, more than anything, along with an array of other samples from other stops. Such an idea seems completely contrary to the concept of Andromeda, however. So I suppose if it were from the outside then you'd have to admit that the species of this galaxy hijacked it. Going the easy route those on board are dead, and those that found it figured out that there was some stasis issue that they've now solved.
The hard route is that they found them and killed their occupants, of course.
Then there's the idea of a super relay, that brings about its own plot problems, with the Reapers specifically . . . meh.
None of it really fits for me.