I can't do the hard-science for your other two questions, so I'll try to just focus on this one.
keep the whole thing running smoothly
Note: I've made a ton of estimates and assumptions in this answer. There are obviously many different ways to tackle this problem, and this is only one. There are also many, many other considerations to make in regards to the scenario, but I have indiscriminately hand-waved them away for now.
First of all, let's try to figure out how much storage space you'll need to hold your civilization. Estimates for how much a human brain can hold vary from 1 terabyte to 2.5 petabytes. Let's just take the high end of that estimate and work with that.
You're going to want some kind of redundancy as well. There are different strategies for redundancy, especially considering these are people we're talking about, but for simplicity, let's go with a RAID 10 approach.
$$
2.5\ petabytes * 4\ drives * 3\ 000\ 000\ brains = 30\ zettabytes
$$
Now let's consider what medium of cold storage to use. Unless you specify otherwise, there should be no reason why these brains need to be conscious during the trip. From the passenger's perspective, the passage of time should be imperceptible from the time their consciousness is loaded into storage, to the time their new bodies are created. So let's use magnetic tape for our long-term storage. In my opinion, there really is no better way to keep things running smoothly for your scenario.
The latest generation of LTO Ultrium (the standard form factor of Linear Tape-Open magnetic tape drives), LTO-6, has a raw data capacity of $2.5\ terabytes$. Currently, future generations of LTO are already planned, all the way up to LTO-10. So, let's go with LTO-10 as our storage medium.
LTO-10 has a planned raw capacity of $48\ TB$. At a compression ratio of $2.5:1$, we should be able to store $120\ TB$ on one tape.
$$
\frac{30\ ZB\ of\ storage\ needed}{120\ TB\ per\ tape} = 250\ 000\ 000\ tapes
$$
Using this product's specifications as the basis for what a future-gen's specs might be, the mass of all those tapes would be $250\ 000\ 000 * 0.27\ kg = 67\ 500\ 000\ kg$, or about times the mass of the Titanic. The volume for all those tapes would be $11.3\ cm * 2.79\ cm * 11.1\ cm * 250\ 000\ 000 = 87487\ m^3$, or roughly 0.4 times the volume of the Hindenburg. As of writing this, you can buy this LTO-6 tape at 10 for 320.00 USD, or $\$32.00 * 250\ 000\ 000 = \$8\ billion$, or the value of Spotify. According to these numbers, I think your Super Star Destroyer-size ship is probably a little excessive.
Of course, your ship and its payload will have to be maintained and piloted somehow. I don't know if you want to allow robots to perform the routine maintenance and piloting, or have some small group of humans with bodies awake throughout the journey...that would add a layer of complexity to your ship design, for sure. Or you could have some ungodly combination, where you have a few human consciousness' plugged directly into the shipboard controls and sensors, able to maintain the ship. That actually sounds like a pretty awesome plot setup.
The good thing about these tapes is that they have very easy requirements. It looks like the ship library they are to be stored in should be temperature- and humidity-controlled at ranges between 5° to 23° C, and 20 to 60%, respectively. And that's about it. The rest of your ship and its energy is dedicated to just being a ship.
Let's say we use these bad boys for our data library. Of course, we'd probably have a custom build in our ship, but I'm just using these as a basis for comparison of specs. Also, I'm assuming data transfer rates are static over the next 50 years. We'll need $2\ 500\ 000 \div 12\ 006 = 20\ 823$ of them to hold all our brain tapes. The bad thing: these are probably crazy expensive. I couldn't find a number, but I can only imagine. The good thing is the R/W speed. If we were to use a single drive to read out all of the tapes it would take:
$$
\frac{30\ ZB}{400\ MBps} = 2\ 377\ 000\ years
$$
Since we were RAID 10 redundant, we can go ahead and cut this number in half on the read side, when we get to our new home planet. In order to cut the total read/write times to within 1 to 2 years, we'd need about 1.5 million tape decks. At about \$5,500 per drive, they would cost us about another \$8 billion.
But with our fancy tape libraries and reading decks, we would have all the reading done in
$$
\frac{30\ ZB}{138\ TB/hr * 20\ 823\ libraries} = 435\ days
$$
Not too bad, considering.