I can give you a proof for (1). Use manhattan distance. Take an arbitrary point P. Sand can not get arbitrarily far from P.[0] Thus there is a highest distance

I can give you a proof for (1). Use manhattan distance. Take an arbitrary point P. Sand can not get arbitrarily far from P.[0] Thus there is a highest distance n from P sand can reach. Whenever sand collapses, at least one grain gets further from P. Thus there are no collapses at distance n. There are a finite number of points at distance n, so there are a finite number of collapses at n-1. But that means a finite number of grains enter n-1, so finite number of collapses at n-2. And so on. [0] Consider the path that the sand would travel to get super far out. The path cannot contain any adjacent empty cells (proved in the article). Thus it would take infinite sand to get infinitely far out.