Thresholds for Infinite Expected Cluster Size/Diameter

Question

In a many infinite random graphs, such as lattices, there is a non-trivial threshold $p_c$ above which, if we independently and identically select sites or bonds to belong to a set, we will obtain an infinite connected cluster all of whose sites or bonds respectively belong to the set. This implies non-trivial respective thresholds at which the expected size and diameter of the cluster containing a site becomes infinite. Are these in general equal to $p_c$? If so, will infinite values arrive at $p_c$, in $(p_c,1]$, or would it depend on the graph? If we choose our graph to be the rooted directed tree in which every vertex has countably infinitely many out-neighbours and weaken our i.i.d. condition so that only the sets of outneighbours are i.i.d., then we have a branching process. Diameter is the maximum eccentricity and the eccentricity of the root is just the length of time before extinction. We can show using Markov's Inequality that when the expected number of offspring is less than $1$, the expected extinction time is finite, and since the probability of extinction is less than $1$ when the expected number of offspring is greater than $1$, in this case it is infinite. So a related question is whether the expected extinction time is infinite when the expected number of offspring in a branching process is $1$.