Don't be worried about asking a professor if you can participate in their research with them. You're correct in thinking that research is a huge part of the grad school process, so the earlier start you have the better chance you will have at getting into the program you want. Even if you...
I can only speak to your second question, and from a physics major's perspective rather than EE. That being said I know that research is a huge part of getting admitted to a graduate school (at least in physics). Some would say its the most important factor actually. So yes -- start as early...