Ships in the universe of Iliad-07 travel via Constant Accelleration. Time on the ship slows down, allowing vessels to span the gap between stars, or even potentially other Galaxies, in decades. However, due to Relativity, much more time passes on Earth and the Colonies than passes on the ships. In the case of Iliad-07's trip to 18 Scorpii, the trip for the ship is 17.8 years ship time, 96.1 years Earth time.
Colonies have to be able to sustain themselves for decades, or even centuries. This puts a lower limit on the number of crew for a ship. You enough people to be able to spawn enough people to run the ship for several generations without inbreeding. And to account for accidents, people out on maternity/paternity leave, and the fact that people get old and are going to retire at some point.
Another important factor is Minimum viable population. For vertebrates (and we are vertebrates), geneticists use the 50/500 rule. You could keep the lights on with 50 people for a single generation, but you need at least 500 to prevent genetic drift. The block three Iliads start with a population of around 750, because we also have to account for people who may not have the interest or ability to have children.
But, how do you select those 750 people? Randomly. Or as near a random process as possible. Ideally you would select people in their 20s. They are educated, but haven't put down much in the way of roots. Any children they have are still small and can acclimate to the culture on board. They also have all of their productive working years ahead of them.
Genetic diversity isn't only a problem for the people on board. Minimum Viable Population applies to livestock and pets.
The vessel leaves Earth with actually only start with 380 or so cows. (To economize shipping.) But ship also has thousands of ampules of bull sperm, cryogenically frozen. The sperm can keep for decades, and allows the farmers on the ship to "top off" diversity at regular intervals. Rather than having strictly dairy cows and beef cows, the breed selected would probably be one that could serve in both roles. For the moment I have Dexter Cattle pencilled in.
I suspect there will also be a desire to have pets. For cats, dogs, hamsters, birds, ornamental fish, and so on the same 50/500 rule applies. There would be less emphasis on breed, and more on taking aboard individuals that can interbreed and have a somewhat healthy bloodline.
The Pariah Dog from India is a naturally evolved breed of dog. They are intelligent, and very, very hardy.
For cats, the best bet may be to actually collect strays from around the world.