As for the ship and being loaded on and what not:
Stasis pods or whatever you want to call them are located on the ship, one would typically be frozen once they arrived on the ship as it's cheaper and safer. Beside, someone has to pilot the ship until autopilot kicks in. I do not have a specific model to give for the ship as I think of it more as a space station, but instead of it being in orbit as its main function, its main function was travel. It will serves as a traditional space station until a home colony is founded. If no colony site can be established it can be a permanent home but that is not the primary goal of the project, think of it as a fail safe.
There are smaller "shuttles" or "cruisers" that can be used for transport from the Mayflower to any planet being scouted.