I would drop the notion that your technology includes artificial gravity (beyond spinning) and engines that can get you to half the speed of light. Which as many point out limits time for the generations and leads to the problem of hitting stuff.
I would assume that a spinning cylinder of a decent size with the interior designed to provide space, light and "nature" with the center axis being a source of light and "rain" and your vessel has decks below (outside of the inner space) that provide space for farming, then industrial tools, then storage and lastly the outermost deck would be flooded with water not only to protect the occupants from cosmic radiation, but also to make certain that the ship has enough water to make it across the vast expanse of interstellar space.
I would increase crew/passenger size to 100,000 with an expected expansion to 300,000. I would also consider that the on board time between stars is about the same as the remainder lifespan of a 20-something first generation person. 50 to 60 years seems reasonable thus you would have grandparents who remember green hills, blue skies and a horizon that bent down instead of up.
10% light speed tops makes a reasonable on board trip time. I'd go with something along ion engines, low thrust that continues for a very long time. Instead of turning the ship you would have thrusters that would point bow-ward at an angle, thus your deceleration time would need to be longer than your acceleration time. This way you can use both ice(s) plow/shield and a Bussard Ram Scoop collector to provide more fuel.
Ion thrusts are weak - 5.4 Newtons of thrust is the latest and most powerful ion engine. One can convert newtons to gravity... But I'm not doing it here. 947
I can pretty much tell you that you would be pulling .01 g which would hardly be felt and the spinning craft would have a greater force on people than the continual thrust.
Your ship could get up to a decent velocity given lots of time, but a half the speed of light will bring way too many other technical questions to mind as previously explored.
300,000 colonist/terraformer/ship builders arriving in a stellar system is a good start. One could argue that the ship is feeling crowded by this time so passengers and crew would be hungry for more space. And humans are real good at making new humans so 20-40 years at this new star system, given enough space, humans could be at the half million mark plus.
They would need to stay in that system long enough to make enough humans to fill two ships... plus leave behind enough colonists to make a new world, make more ships - what ever direction they want. figure another three generations of time (say 60 years) and 1.5 kids per parent - magically the human population is at or slightly above one million. To give you an idea of how fast humans can make more humans, just look at population growth from 1900 (1 billion humans) to 2000 (7 billion humans).
So yes, your basic idea makes sense. Humans making more humans, the desire to find new fresh untouched worlds to explore. Perhaps groups of like minded people are off to make their own worlds by their own rules...
As for size. Rendezvous with Rama might be helpful. Not only with size, but with interior design to handle space travel. 947