First off, as shown by WALL-E's behavior in the final scenes of the movie, most WALL-Es were not self-aware, and were relentlessly single-minded, managed by human teams as part of Operation Cleanup. The idea, laid out by Andrew Stanton in the commentary, was to set them loose over a wide area, and they'd divide and conquer, gathering up the trash and building the skyscrapers of garbage, which would then be consumed by the big conveyor-belt eating machines and incinerated. However, burning all of that trash had unintended consequences ("rising toxicity levels" as mentioned in the A113 message), and eventually the remaining humans who were managing Operation Cleanup couldn't survive the conditions, and had to leave in a hurry. The robots, it's implied, were never actually turned off; they were just abandoned to their fate by the humans, who assumed all the robots would eventually break down for lack of maintenance.
The one WALL-E left, the title character, has somehow gained digital sentience (and a personality), through some programming abnormality, a fortunate accident, or simply the slow accumulation of knowledge over the 700 years he's been functional. How and when it happened is unknown, but it's implied to make WALL-E unique. By becoming self-aware and able to learn, he has also gained the smarts to stay alive (thus furthering his primary directive) by learning to cannibalize the other WALL-Es as needed. If we accept that WALL-E's sentience and ability to cannibalize is unique among his kind, which it is for all we can tell, then it follows naturally that WALL-E's the only one of his kind still functional, because all others, unable to adapt, would have succumbed to the worsening conditions on the planet over time. The opening scenes show that WALL-E has taken this trait one step further, collecting various things he's found interesting among the refuse.
Andrew Stanton and his art team took pains to contrast WALL-E and EVE heavily in form and function. You pretty much understand at a glance how WALL-E works; tough, sturdy, no-frills utility bot, powered by electric motors, servos and hydraulics, and what was probably little more than a Roomba's programmed intelligence at first. EVE, as Stanton says, is the Porsche to WALL-E's Jeep; fast, powerful, built to survive alone for months or possibly years, programmed from the start with self-awareness and learning capability, and because she represents the cutting edge of human technology as of shortly after the starliners left, which itself is probably hundreds of years beyond our present time, we don't quite understand exactly how she works.
However, until she's exposed over time to WALL-E's quirky digital personality, she's almost as single-minded about her directive as WALL-E originally would have been; while she shows some human emotions, like happiness, fun, and frustration, all before she's formally introduced to WALL-E, she has a job to do, and she knows it, placing her friendship with WALL-E second (sometimes last) in her list of priorities until much later in the movie.
As far as Earth technology being built to last, not everything left behind is still working; in fact the majority of the tech left behind is non-functional. Most of what does still work does so by convenience for the story. The relatively advanced tech that we see still working is primarily in the opening scenes, as a way to tell the audience what's happened to the planet and its former inhabitants, and in the final scenes to allow the docking and unloading of the Axiom from a docking cradle that would have been immobile and idle for 700 years. Most other Earthbound technology that still functions does so to make a gag work, like the singing fish, the iPod and VCR, or the CRT TV and Pong game.
There's plenty that isn't explicitly broken down for the audience, but for what it's worth I think WALL-E's universe is among the more well-designed and well-explained of sci-fi alternate realities, and it does so with precious few words.
