1, Cryogenics/ keeping a population in a suspended state, if you're moving an entire population like that it'll take a lot less space to just stack them in chambers than it would to say include normal food, medicine, exercise and entertainment.
I don't know if they're giving size numbers in GCIII yet (or if they will at all), but in GCII TotA you could look up two of the dimensions and the mass of the ship (Ship Details -> Intelligence Report). A default GCII TotA colony ship had listed dimensions of 933m and 516m and massed 1.6 million metric tons. If you assume that the ship was 33m x 516m x 516m, its volume was about 250 million cubic meters. A coffin is roughly 2m x 1m x 0.5m, and so you can fit about 250 million coffins into that ship; this should serve as a rough estimate of how many people you could fit onto the vessel if you required a negligible amount of space for drive systems, power systems, navigational systems, life support, etc. Alternatively, we could use the listed mass of the ship and an assumption about the construction (e.g. iron construction, 90% hollow) to estimate the internal volume of the vessel to then estimate the passenger capacity (under this set of assumptions, we find that the ship's volume is ~2e6 m^3 rather than ~2e8 m^3, and so we expect that the ship can carry about two orders of magnitude less people than under the assumption of a 933m x 516m x 516m ship). One further alternative would be to assume that essentially 100% of the ship's mass is due to the people carried; if we assume that an 'average' person has a mass of 90kg (~200lbm), then a ship with a mass of 1.6 million metric tons can carry about 18 million people (1.6 million metric tons = 1.6 billion kg ~= 18 million 90kg bundles).
Alternatively, we can look at real-world passenger vessels and estimate based off of those how many people would fit. Using Wikipedia's articles on the RMS Titanic, RMS Queen Mary, and RMS Queen Mary 2, we can come up with about 63,000 to 110,000 passengers (assuming displacement in long tons because 1 long ton ~= 1 metric ton). If we instead use Wikipedia's page on the Boeing 747 and use the numbers it provides for the 747-100B (550 passengers, and assume the minimum operating weight rather than the probably more reasonable maximum takeoff weight), we can push the number of passengers up to about 35 million.
However you cut it, if GCIII colony ships are approximately the same size as GCII colony ships, then the number of people the ships can carry are rather excessive even before you consider that you're probably carrying around some heavy equipment to help get the colony started and require some amount of space for the ship's systems.
(2) is not neccessarily absurd, since planets outside our system could be easier to colonize (if you can reach them) if they're are "Earth-like" from the start, which Mars isn't.
It also depends on how much time elapsed between the development of intrastellar flight and interstellar flight. If we're talking about spending millenia more or less trapped within the home system due to lack of superluminal drive technology, then sure, I could see long-term terraforming projects and in-system colonization before more appealing worlds which are too far away to serve as practical colonies. Of course, how far is too far is open to debate; the Drengin are known to have been willing to wait ~75 millenia to invade the Torian homeworld via stargate, so presumably they or another species might be willing to wait something like that amount of time to colonize an ideal potential colony (hey, the Drengin did it; there were already people living there and the databanks describe it as an invasion, but the difference between an invasion and a colonization effort has historically been murky anyways).
1) A single colony ship can transport 3 BILLION population - how does it do that? Frozen sperm and egg cells?
Doesn't really work unless you have the ability to greatly accelerate the growth rate, and you'll probably want some kind of knowledge imprinting technology; children below the age of five are likely in my opinion to be a net drain on the colony's resources, at least if they're similar to humans, and probably still aren't that much of a net boost for the colony until their teenage years. On top of that, unless you have some kind of knowledge imprinting technology, you're looking at a colony full of several billion uneducated people, which means they'll probably be more or less useless as researchers or as workers in the more technical fields (and, for better or worse, you can likely expect that the jobs in less technical fields will continue to disappear as automation becomes more and more prevalent, which means that these people are not really worth much to your empire, as they provide warm bodies but not much else).
