Sorry for not replying earlier, been a busy week. Thanks for the writeup, most helpful. :)

We can also have PM changes without elections - here, the PM is (almost always) the leader of the ruling party. It's possible to have a PM who wasn't elected by the public at all, in fact, if the ruling party's leader didn't win their riding. I'm a poor historian, but I think if that's happened the situation hasn't lasted long.

Our PM isn't actually our head of state though; that's the Governor General, the Queen's representative. Practically speaking, the GG is appointed by the PM and approved by the Queen.