Let me start by saying that you should split question up into its components and ask them in separate threads. Second, all this is going to be pretty subjective, because it all comes down to your imagination - the science can be made up to a point.
As far as the space people might need? It's going to come down to the individual. If you screen your candidates psychologically, they might be able to handle cramped space quite well - look at nuclear submarine crews.
Children adapt to pretty much any environment, and if they grow up under those conditions they're not going to question it - it'll just be the way it is. The challenge most pressure is primarily going to be on that first generation. Years down the line people will think that growing up on a spaceship is absolutely normal
As long as you make sure to maintain population control you should be fine.
Sanity & Health
A green space with a mock-sun, etc would probably be a big plus for everyone, especially in the long run. Also, I should note that lack of gravity, and even reduced gravity, negatively impacts our health over time - check out the effects of spending a long time in space on astronauts. This ship should be able to generate Earth-normal gravity if you want people to actually live a full life-span.
Also, radiation out in the universe might eventually cause mutations in the DNA of the passengers, or even cause infertility. You may want to bring embryos from Earth, on ice, to grow into adults and introduce them into the population over time in order to counteract the effects of in-breeding, etc.
Last but not least, spaceships make great breeding grounds for viruses and bacteria - read up on the situation on the ISS. This ship will need a system to disinfect areas, or the passengers will need to have some kind of heavy duty inoculation to disease - maybe nanites or something?
A long journey
If your engineering and manufacturing capabilities are solid, and as long as no cataclysm befalls the ship, you should be able to keep going for generations and generations.
This ship could stop and capture meteor/comets/asteroids in order to replenish its stock of drinking water, mine for minerals, etc. The one danger when doing this is picking up some bacteria or virus from a long-dead world which might sweep through the population and kill everyone, or turn them into mutants, in the best tradition of sci-fi.
A generation seed ship is 100% imaginable. The engineering doesn't even have to be too out-of-this-world.