Yes, this is correct.The entire critter has to be pushed into empty space, as stated on p. 285 of the PHB.
Yes they can. No more odd IMHO than size having no bearing on pushing and pulling.That seems, well, odd. Can a large creature therefor chose to stand in a place so that only one square is on the ground? That seems odd, and even worse for a huge creature.
The main thing that seems odd to me is that a horse can walk right over a 5-foot-wide chasm without even needing to jump. Best to not think about it too much.