Hopping Rabbit

The Problem

A rabbit starts at position 0 on the integer lattice $\mathbb{Z}$. At each step, it hops forward by an integer amount chosen uniformly from $1,2,3,4,5,6$ (think of it as rolling a die).

1. What is the probability that the rabbit ever lands exactly on position $n$ (for large $n$)?
2. What is the expected number of hops to reach position $n$ or beyond?
3. Generalize: if the hop distribution has mean $\mu$ and the rabbit starts at 0, what is the long-run probability of landing on any specific large $n$?