Well one you can't replicate everyone's exact comfortable climate, I like cold, not everyone does.
True, but for the average human anything between about 74 and 85 degrees is going to be livable. If you like it colder or warmer you could just wear a jacket or a t-shirt.
What happens when people feel there entitled to have what they want, which is common. It doesn't matter if its livable, its about comfortable.
the people on the ship will have curiosity about where the ancestors came from(example: if that weren't the case Ancestry.com wouldn't be here)
True as that is, that curiosity for the most part can be satisfied by a simple trip to the ship's library. Most of US have never visited the place where our ancestors came from either.
Apparently you don't know many people, becuase ancestors doesn't mean that its your 15 grandfather. Your grandfather is your ancestor. Alot of people do actually visit were their ancestors came from. Poeple who had their ancestors in concentration camps back in ww2, have visited that place.
One last point, you have to remember I said what will a future ship look like. So you might not need a generational ship, unless you leave the galaxy to populate another galaxy. So please stop thinking modern, you wont need it to be generational, becuase it would be FUTURISTIC.
If you're not using a generation ship or even a colonial habitat, then it's a non-issue anyway. Anything fast enough (FTL) to make the trip in less than a generation would, by definition, permit easy movement back and forth for anyone who wants to go.
But this assumes that some type of FTL drive is even possible with future technology, which is hardly a given. A generation ship with future technology could easily be an entire asteroid hollowed out and propelled through space with nuclear pulse thrusters, or you could avoid those sorts of complications with cryogenics and the crew stays in hibernation for the 60 to 70 years it takes to get anywhere. Just speculating on the basic premise, it's sort of an open-ended question, and it largely hangs on whether or not you expect FTL drives will be available in the future or ever.
Anyway, "modern" and "generation ship" don't even belong in the same sentence, the technology to attempt such a thing won't exist for another hundred years (i.e. in the future). Something the size of an O'neil cylinder or Babylon-5 with a fusion engine attached would eliminate ALL of the problems of "homesickness," a journey of 30 to 50 years would be more than sustainable for a small but growing population. Children growing up in such a vessel would have no difficulty coping with the voyage, but it might be tricky convincing them to relocate to whatever Earthlike planet they find below.