The statement about percolation is true quite generally, not just for Erdős-Rényi random graphs, but also for the square grid. Above the critical threshold, the giant component is a positive proportion of the graph, and below the critical threshold, all components are finite.
The example I'm thinking about is a non-random graph on the square grid where west/east neighbors are connected and north/south neighbors aren't. Its density is asymptotically right at the critical threshold and could be pushed over by adding additional west/east non-neighbor edges. The connected components are neither finite nor giant.
Another month, another rationality quotes thread. The rules are: