For (2): they are the needed modification of the non-homotpy ones when you want the result to change only by an homotopy equivalence ff you change the input to the construction by homotopy equivalences.
Later: My intuition comes from the following image: say you have a space $X$ on which $\mathbb Z$ acts, and you want to take the quotient. I look at the homotopy version as the result of tacking the orbits: you attach a thread (a copy of $\mathbb R$) to each orbit... If later you want to take the "real" quotient, you just need to pull the threads, and the orbits contract to a point (I know it makes a funny noise when you do this!). When you take homotopy quotients by another group $G$, you need to get yourself a "$G$-shaped thread", which is what we write $EG$.