Question: Right after WALL-E introduces himself to Mary and goes off to find EVE, Mary is surprised to find that the Axiom has a pool. First of all, how could she not have noticed that after living her whole life there? She must be at least 20 years old, and the pool, or Lido Deck, is obviously a very popular place for people to hang out at. And second, how does she even know what a pool is? If she's never been to Earth and she never knew the Axiom had a pool, then she's never been in one. And we can see that the captain is very uneducated about things on Earth. So how does Mary know what a pool is if she's never seen one before?
Question: How could Auto act against his directive? He's playing the top-secret order he got (never to come back to Earth) in front of the captain, yet isn't able to act against it again because things might have changed in the 700 years since the order was received. And in the face of the new evidence (the plant), doesn't that contradict the order?
Answer: One of the key points of the movie is that programming can evolve - WALL-E being the most obvious case in point. After all this time, Auto's entitled to be somewhat erratic in how he deals with things. Besides, he's still very firm on his primary directive, to prevent the return to humanity to Earth - keeping the existence of that directive a secret is rather less critical. As for the plant, that does very little to invalidate the directive. It may show that the principles underlying that directive are flawed, but Auto's not got the leeway to deal with that. The directive still stands.
Question: The only animal surviving is a roach. It survives on wrapped food and shelter that Wall-e provides. There is little to no water in sight but if and when there is water it seems to be very toxic with sludge and acid rain. So how does the roach manage to live with no fresh water?
Answer: The "mere" stun gun seems to have been designed to incapacitate robots (such as the "problem" robots freed by Wall-E earlier), and also lasts considerably longer than a flash of lightning...AUTO shocks Wall-E for several seconds. Besides that, the lightning scene is played for laughs, while the other is dramatic, so some suspension of disbelief is at play.