The probability that a random walk on $Z^d$ from $x$ hits a vertex $y$ is proportional to the Green function $G(x,y)$, which is well known to decay as $c|x-y|^{2-d}$ (using Euclidean distance).

The expected time to hit $y$ conditioned on hitting it at all is of order $|x-y|^2$.

You can find these in any book dealing with random walks, e.g. Spitzer. You can prove these from the local CLT among other methods.

The probability that a random walk on $Z^d$ from $x$ hits a vertex $y$ is proportional to the Green function $G(x,y)$, which is well known to decay as $c|x-y|^{2-d}$ (using Euclidean distance).

The expected time to hit $y$ conditioned on hitting it at all is of order $|x-y|^2$. One way to see this is to compute $\sum_n n p^n_{xy}$, which using the local CLT is of order $\sum n^{1-d/2} e^{-|x-y|^2/2n} \approx |x-y|^{4-d}$. Divide by $G(x,y)$ to get the expected time to hit conditioned on hitting $y$. (Subsequent hits do not have a significant effect.)

You can find these in any book dealing with random walks, e.g. Spitzer. You can prove these from the local CLT among other methods.