Do they actually build stuff or just wire them? I know this is probably a stupid question.
no it isn't, you learn a lot other than just wire handling here is a link of what you learn in a 4 year course of bachelors in engineering csvtu.ac.in/syllabus_new.htm
No, you also have to know about electricity. Even as an electrician. Please don't take this career up, you'll be a crispy critter before long.
It certainly does not. There is a big difference between an electrical engineer and an electrician. An electrical engineer may well have a 4 year degree course to complete. Such a course will have a high Mathematical content. Electricians do not follow such a rigorous course and will be more hands on. Not a stupid question !
Electrical engineering is a broad and expansive field that covers almost every aspect of any device or piece of equipment that uses electricity. I have a friend who is a electrical engineer (and is also the smartest person I know), who knows how to design complex Printed Circuit Boards, right down to complex wiring jobs. He currently works as a product engineer for HP. But they do a lot more than that, from designing electronic devices, to working with and designing the high voltage equipment that is used in power grids. Pretty much anything that uses electricity was designed or is maintained by an electrical engineer.