To answer your first question (regarding existence of the limit): I never remember references, but all you have to do here is show that for any two far enough starting points, there is a coupling of the RWs started at them, such that with high probability the two paths hit the aggregate at the same point (for example, because they start walking together before hitting the aggregate). In $\mathbb{Z}^2$ it's pretty straightforward to do: if you let one walker walk till it hits the aggregate, then with high probability its path will separate the aggregate from the starting point of the second walker. Then you let the second walker walk till it hits the first path and follow this path thereafter.

As for the second question: as I said, I never remember references, but I believe that you can find how to calculate the harmonic measure exactly using the 2d potential kernel in Spitzer's "Principles of Random Walks".

To elaborate on George's comment and your reply (again, I'm pretty sure all this appears in Spitzer): Let $A$ be a finite set of vertices and start a SRW at $X_0=x$ and for some $y\in A$ we look at $\mathbb{P}(X_{\tau_A}=y)$, where $\tau_A$ is the hitting time of $A$. Then this probability is equal to $\sum_w \mathbb{P}(w)$ where the sum is over all paths starting at $x$ and ending at $y$ and not going through $A$. Since the SRW on $\mathbb{Z}^2$ is reversible this is equal to the sum over paths strating at $y$ and ending at $x$ and not going through $A$. This is exactly the expected number of visits to $x$ for a SRW started at $y$ and killed upon returning to $A$. This is proportional to the probability of hitting $x$ before returning to $A$ (again, when starting at $y$). If we take $x$ to be far away we see that conditioning on hitting $x$ before returning to $A$ is the same (asymptotically) as conditioning on the walk not returning to $A$ for a long time.

More can be said about the distribution of the conditioned RW, but right now I have to go.