I can totally see that people who get 100% service from robots for 700 years on an isolated cruise ship would eventually breed total and complete dependence.
What is not believable about that?
I saw that as the moral of the story.
Global warming was just the reason people left the planet, not the lesson of the story. It could have been a plague or sun spots that forced the people to leave Earth and the same lesson would still be there. If this story was pushing the global warming theory the people would be abusing the ship the same way they allegedly abused the planet.
They are abusing the ship, none of the people realize it though. As far as the people are concerned, everything is sanitized, prepackaged, and handed to them via robots. They still produce garbage up the whazzoo and recycling isn't exactly high on the priority list. You see evidence of this when Wall-e and Eve get dropped down the garbage chute and meet the two Wall-r's shoving huge cubes of trash into deep space.
They don't necessarily explain how the ship can sustain itself for 700 years in deep space when they're shoving scarce resources out the ship. I guess the assumption is they get raw materials in some manner. In any case, like you said, it's not really the underlying message in the movie.
They didn't explain if WALL-e was given a personality by accident(like Johny-5) or if he was programmed with one. Eva and the other robots had a personality so I'd assume he was programmed with one too. It would have been better as a Robinson Crusoe type story. Focused on him "developing" a personality and figuring out that he was alone and then trying to find other forms of intelligent life.
Interestingly, I suspect he was given one by accident. Evidenced by the destuction and replacement of one of his circuit boards. After his "death" he doesn't express any personality until Eve holds his hand. I get the impression that each of the robots have the potential for a personality, but something needs to trigger it. The stewards all lack personality, yet each of the robots in recovery exhibit a personality. Eves develope over time. The secretary expresses nothing until it encounters Wall-e. So on and so forth.