The trailer tells us nothing, we just see it drifting through space with nothing to compare it to. Dreadnaughts are massive, with thousands of crewmembers. It might need to be bigger than a dreadnaught, but not not by much. They wouldn't need tens of thousands of each species. A few thousand would be more than enough to restart the population. Besides, if their traveling in stasis pods, they could be packed tightly and wouldn't need to wake everyone up until they found a world viable for settlement. The Collector ship had millions of pods inside with lots of empty space. So there goes that problem.
The thousands of passengers isn't going to be the only thing the ship will carry it has to carry the necessary resources to set up several colonies for the various different species. The ship also has to be capable of dealing with the stress of traveling in FTL non-stop for several centuries and no such technology existing during the time period of the trilogy. There is also the fuel needed to carry out the journey which is going to be extremely massive for a journey that's going to last centuries. All of these factors means this is going to be a large scale feat that'll take years and years of preparation before the journey can even start.
The Alliance didn't trust Shepard while she was fighting the Collectors, plus they probably didn't know it was a Collector ship at all, just that it was advanced. That's why I said it was derelict. There were no collectors aboard, it was a dead husk.
Cerberus would've known if the Alliance had found a collector ship since there would've been plenty of Cerberus sympathizers and operatives in the Alliance.
The Alliance has heard of them since they joined the galactic community, but like almost everyone else have written them off as superstition. At least that's how it looked to the public. It wouldn't be beyond reason to think the Alliance lied about them, since they lied about the Reapers until the Reapers introduced themselves to humanity at the beginning of the Reaper War.
The collector's are so rare that many in the galaxy consider them a myth it's unlikely they would lie to the public about them if they've yet to encounter them themselves.
There are more than two. Many more. There is the one encountered and destroyed on Fehl Prime, there is the one that is destroyed when Shepard attacks the Collector Base, and the Reapers called in all the rest during the Reaper War. Nowhere is it suggested only two were operating in the galaxy before then.
The collectors seem to be sort of a spec-ops unit for the reaper's and only have a handful of ships the reaper's would've gone to great lengths to keep them out of major engagements.
Collector technology has never shown any signs of indoctrination. Chances are it was Prothean tech that was upgraded by Reaper tech, like how EDI was human tech upgraded by parts of Sovereign, yet she never indoctrinated us.
The collector's were known to use reaper tech, they help build the reaper's themselves so its safe to say collector ships do have reaper tech.
Plus the Leviathan DLC establishes that there is Anti-Indoctrination shielding available since it is used on a piece of Sovereign, so they could use that as well.
A small reaper fragment is one thing an entire ship is something else entirely, it'll require large quantities of power to keep the shield operational fr centuries. Needing to shield the ship from indoctrination will only add to the problems with powering the ship itself.
As for making sure the ship is spaceworthy, that's a simple enough thing to do. Wouldn't even take much resources to refurbish and upgrade one to suit us better.
Even modifying and making sure it won't indoctrinate the crew will still take a significant amount of time, this project will be a missive undertaking no matter what they do.