Random structures: walks, percolation and random geometry

How many of the $4^4 = 256$ walks of length $4$ on $\Z^2$ (steps $(\pm1, 0)$, $(0, \pm 1)$) end at the origin?

On which of these graphs is simple random walk recurrent? (Choose all that apply.)

In the rooted binary tree (root of degree $2$, all other vertices of degree $3$), what is the probability that simple random walk started at the root never returns to the root?

How many edges of $\Z^2$ join a vertex on the boundary of the square $[-2, 2]^2$ to a vertex on the boundary of $[-3, 3]^2$? (This is the cutset $\Pi_2$ in the Nash-Williams argument.)

In Peierls' proof that $p_c(\Z^2) \ge \frac13$, which fact justifies the bound $4 \cdot 3^{n-1}$ on the number of self-avoiding paths of length $n$ from the origin?

In $G(n, c/n)$ with $c = 2$, roughly what fraction of the vertices lie in the largest component when $n$ is large?

Which Lean statement says that the expected number of returns to the origin of simple random walk on $\Z$, $\sum_n \binom{2n}{n}4^{-n}$, is infinite?

Which of these are established theorems (not conjectures or numerical observations)? (Choose all that apply.)

Write return_prob_1d(n) returning $P(S_{2n} = 0) = \binom{2n}{n}/4^n$ for simple random walk on $\Z$ as an exact Fraction, return_prob_2d(n) returning the probability for $\Z^2$ (use the rotation trick), and expected_visits(f, N) returning $\sum_{n=0}^{N} f(n)$. The tests check exact values and the slow growth of the 2D sum.

Simple random walk on $\Z$ returns to $0$ with probability $1$. What is the expected time $E[T]$ of the first return?