Drunk Man

The Problem

A drunk man stands at position 0 on the integer lattice $\mathbb{Z}$. At each step he moves left or right with equal probability $\frac{1}{2}$.

1. What is the probability he ever returns to the origin?
2. What is the expected time of his first return to the origin (conditioned on returning)?
3. What changes in dimensions $d=2$ (drunk on ${\mathbb{Z}}^2$) and $d=3$ (drunk on ${\mathbb{Z}}^3$)?