There are a lot of problems with an Ark. Alot of it would need some sort of explanation.
I could try answering some of these for fun and profit.
-Why don't they turn around once the Reapers are defeated? They would know thanks to Quantum entanglement communicators and unless the speed is staggering they would almost certainly be much closer to the milky way than Andromeda.
These are the people so scared of the Reapers and so convinced this is the end of everything that they've decided to abandon the galaxy, it's entirely possible that they don't have any channels of communication to the rest of the galaxy because they're so afraid the Reapers will be able to track it.
-If it is technologically so easy to cross a galaxy, why haven't the Reapers gone before? Or if its simply because of their programming (which is a lame excuse) what is stopping them once god shep takes over or everyone becomes synthesized? Having that level of technology is a glaringly obvious writing mistake if you want to keep the galaxies separate.
As far as we know the Reapers had no reason to do so. Their imperative is to preserve life with the caveat of only preserving life within THIS galaxy. Possibly because the leviathans hadn't gone further and thus weren't interested in programming their extermination cycle for a universal scale.
-It doesn't make sense to take a lot of different species. Why would you take a Krogan colony on such a mission? Same for Volus, Vorcha, Quarians. Places on such an Ark would be at a absolute premium and the best chance of survival is to take as few a species as possible. Especially since it needs to be cloaked in secrecy.
No idea actually, species like the krogan make especially little sense since they 'fight with others of their kind in enclosed spaces' which would be a needless resource drain. Planning for quarian suit ruptures would be a pain in the ass too.
-Why Andromeda? It's romantic and they want to set it there, but why would they choose to go there in a controlled decision as opposed Cannis Major Dwarf which is so much closer?
That one's easy, no one wants to live in cannabis dwarf.
-Why are their N7 operatives on board? They are needed to fight the Reapers. They are not needed on a colony mission.
These people are abandoning ship, they're not gonna throw good resources after bad. I'm sure they're aware of the fact that they'll need dangerous people like N7 operatives wherever they're going too.
-Why would we want to play as the cowards (or their descendants) who ran away?
I asked myself that very question, but I think it depends entirely on how it's presented. We COULD be cowards. We could also be the edgy people who will do whatever is necessary to survive. A lot of it might depend on whether or not we ever get contact from the people of the Milky Way again/within the games time frame. It might also be thematically interesting, dealing with the fact that we're the cowards.
-How was there such a technological leap? (I'm not too worried about this, but it will probably have to be explained and I doubt it would be that convincing)
Doesn't necessarily have to be much of technological leap if something like the Black Ark theory is correct.
-Where do they get all the resources from (like the above, so much effort is going into the crucible described as the biggest project in history. Again, I'm not too concerned about this but I doubt it will come across as convincing)
Not sure. If it's an Alliance planet they'd probably notice if the colony isn't giving as much resources as it should to the Crucible project. Maybe it was a rich colony? Maybe they faked their death somehow? Maybe it's Taevor and everyone assumes they killed themselves by setting off nuclear bombs. I dunno.
I think there are some other reasons that I thought of before but can't recall them off the top of my head. It's why I think an accident is such a better choice. As the crucible fires a giant wormhole opens up swallowing a bunch of ships leaving them stranded in Andromeda. The beauty is the simplicity. It doesn't leave a string of story headaches for the writers to have to constantly work around.
Maybe but that brings up a lot of other questions too. If it's a mixed military fleet it's probably the fleet that went out to fight the Reapers right? Would these ships have enough people to sustain a viable population? Would they have the technology to sustain this group of people for however long it may take them to find food/water/fuel again? Would military personnel have the necessary know-how to establish a functional colony for humanity? Why would a military fleet have the technology necessary for surveying undiscovered planets? Would the emphasis be on 'finding a new home for humanity' or 'finding a way to get back to humanity?' etc.