First or all, why would one create a robotic probe designed to search out for signs of life, and then give it a massive arm cannon and a predisposition to shoot anything that moves? Sure it will mainly be firing at animal species and it's designed to search for plant species, but who knows what will be taken out in the collateral damage? And gods forbid that perhaps there's been some serious mutation and the plants have started walking about :) Not to mention that if such a probe found animal life then instead of shooting it it really ought to be quietly following it about to see what it eats. It really makes no sense at all, but it gets a pass because it leads to so many amusing circumstances and images. It also helps a little bit that Eve pulls out the big guns before we find out exactly what her mission is.
Also, how does WALL-E get power to his little cargo container/vehicle thing? I didn't notice any obvious solar panels on top, and after a couple hundred years i expect any batteries left lying about would have naturally discharged. Is he getting the energy from a power plant? Duh, come to think of it where were all those big holographic signs getting their power from? Clearly there must be some power plant still going. Was it so automated (including possibly fuel resupply?) that it's still going after hundreds of years? That seems somewhat implausible without evidence of other robot maintenance bots still being around somewhere. They might stay inside the power plant but it seems somewhat unlikely that WALL-E wouldn't have tracked them down given how curious he was about everything and how desperate he was for companionship. I suppose it's also possible that WALL-E found a busted power plant himself and repaired it, but that seems like a rather big job for a single robot.
Next, as we're leaving earth we see that WALL-E was originally from somewhere in North America. It's hard to tell for sure but it looks like the other continents are rather barren too. Did Africa and South America and other such areas get fully modernized in time to get sucked down by the final collapse? Or was the ecological damage from first world countries so bad that it managed to spill over into the second and third world? For that matter was the issue global warming? Pollution? Something else? A combination? This falls under the category of "there could easily be a plausible explanation they just didn't choose to tell us, but i'm curious to know what it was."
So there was only one of cruise liner (Axiom?) ever built? That seems a little odd, especially if they were planning on it being the last refuge of mankind. You'd think there would have been millions or possibly even billions of people willing to pay for tickets so financing shouldn't have been an issue, and that's overlooking the issue that there should have been multiple governments and possible the UN backing the scheme. And really, if conditions on earth were getting that bad and they had the technology to build a cruise ship capable of surviving on its own for 700 years they should have been able to build, for example, a moon or asteroid colony and send up volunteers who were willing to work rather than pay?
And really, if only a few thousand(?) people went up on the cruise ship what happened to everyone else? Presumably they just died off, but why, given that they had the ability to build space habitats that could last at least 700 years? How hard would it be to build the same thing but put it under a dome, or in a cave, etc? Instead they just sat around and died? All of them? Even the incredibly rich and powerful people who would have had a clue what was going on before mass panic and war broke out? Including, just for example, the president of the US? He was clearly designed to appear incompetent, but even incompetent leaders (or their staffs) are usually pretty good about devising bolt holes for their own safety when necessary.
As for the ship itself, how _did_ it survive for 700 years? Did it have 700+ years of fissionable material aboard? Or some kind of fusion? Or did it not spend all its time lurking in gas clouds as it first appeared and have well disguised solar panels on it somewhere? More importantly, what was it doing for resources? Clearly they were ditching thousands of tons of trash on a regular basis, what were they doing to replace the materials? Did it have a fleet of robotic drones out harvesting resources from asteroids and comets or something? And how could that possibly be more efficient than recycling the waste instead of ditching it overboard? I suppose it fits with the theme of a wasteful consumer culture, but if it was intentional the point was probably too subtle for most of the audience.
And finally, once they get back to earth how the hell do they survive? Well really surviving isn't a problem, presumably the systems on the ship are still running so they can duck back inside for food and shelter and such. However we are shown exactly one plant (which the credits imply will turn into a tree) and one cockroach, and that's it. Given that restoring earth was apparently part of the original plan it's conceivable that there are seeds and frozen embryos on board the ship, but even so with the current conditions on earth how do they get things started again? The regular giant windstorms must make growing crops rather difficult. Certainly not impossible problems to overcome, but exactly the kind of thing i'd like to read about the process of overcoming in a novel :)
Also, as a minor note, when rewatching the movie i need to remember to keep an eye out to see at what point, if any, it would have been plausible for the gopher robot to steal the plant.