I read a paper once that said the effects of an EMP from a nuclear bomb were grossly overestimated. The actual EMP radius is apparently less than the damage radius of the bomb, so it's plausible that electronic robots could actually survive a nuke better than humans: the bomb probably would be more efficient at killing people. Not to mention, once the bomb has blown up, the robots need not to worry anymore, so if the explosion could be predicted, they may be able to hide underground. That, and we know that electronics can be shielded from EM radiation by the use of metal casings or faraday cages, which aren't extremely complex technology (and only the most sensitive electronic components really need to be protected). But who's to say any bombs would be launched at all? After all, if robots were present enough to take over the world, it's plausible there would be alot of them in the military forces. But even if they were not, robots may not be separated from humans geographically, and so the military leaders may have no clear target to fire at that wouldn't result in killing many of their own people.
As for deriving power.... They would be using the same means of electricity generation we have. Nuclear reactors, burning gas, burning coal even, as they may not need to worry about atmospheric pollution and there is enough coal for about 300 years of extensive use. They could even use methanol fuel cells, which can be quickly recharged. I don't think this is a major issue, because if the robots were in a position such that they are everywhere and can control the world, it's a fair assumption that electrical power would be widely available and easy to obtain for robots. If robots do take over the world, we will have handed it to them, and they will already have all the physical resources they need.
As for the question of if/how and why, I believe that humans are much more a menace to humans than robots will ever be. It's clear that when designing robots, they are conceived to be efficient at their task. Why give emotions or goals and self-awareness to a mining/garbage collecting robot? But anyways, assuming everyone had robotized helpers that did everything they wanted, I think the main menace would be our own lazyness. People would delegate all physical tasks to robot, and then the most basic intellectual tasks, and then the less basic, until humans are only working in positions of corporate ownership and administration, and spend most of their life in leasure.
At this point, if robots decided to "take control", it is likely that people would oppose, and that the robots would use force. Whether they would eliminate humanity or not depends on their programming and goals, their interpretation of the situation and their feelings. If they consider humans a nuisance, it is possible that they would find it more practical to eliminate them. If they had attachment to humans, they may simply do this as a passive rebellion/protest.
However, I suspect that if robots were self-aware, they may not be as united as people imagine they would be. There could be hundreds of different models, millions of different robotic individuals. Some may be friends with humans (or even lovers, who knows?), each robot may have its own goals, and the idea of a worldwide takeover may simply not appeal to them, so its possible many robots would actually league with the humans.