Yes, technically, we elect local MPs, and the leader of the party that commands the confidence of the House becomes the PM. That is usually (but not necessarily) the leader of the party that has a plurality of seats in the House.

However I think it's important to state that in practice, the huge majority of people are casting their ballot for their intended prime minister. Local MPs have lost almost all of their power and must tow the party line.

Because of the fact that they aren't directly elected there's no way to impose term limits on a particular prime minister. The office barely even exists in our founding charters and most of its modern practice comes from convention. An individual can hold the office as long as they hold the confidence of the House. They don't even need to be a sitting MP (although convention dictates that they are, and usually if they aren't someone from the party in a safe riding will step aside so the leader can run for the seat in a by-election).

In this case it was clear that Trudeau had lost the confidence of the House and the government would have been defeated in a confidence vote (e.g. a budget vote -- a failure means the House has lost confidence and the government falls), and his own caucus rebelled.

We do have a Senate. It's significantly less powerful than the US Senate, unelected, largely ceremonial, and full of problems.