Basically, registering for a party allows you to vote in that state's primary election.
We don't have to register for a party here, EVER. All Canadians do is register with Elections Canada (for a federal election) or Elections Nova Scotia (or the applicable province) for provincial elections.
The Federal list is compiled using the census - when an election is forthcoming, any Canadian can call their local "Returning Officer" to find out if they are registered at the proper address. Then, you go to the polling station with your lease or a power bill (proof of address), and can vote. No Photo ID required.
Is it possible for the PM to receive less than a majority of his party?
The results are tabulated much the same, again, as Joe Grundy said it is done in England. The country is divided up into 'ridings' - the party with the majority of ridings, is the ruling party. The Prime Minister must run and get elected in her/his riding, or else she/he will have to have one of the elected Members of Parliament step down and try to have that seat won in a bi-election.
Who actually sets public policy?
Any MP/party can put forward a bill to Parliament. It is debated and voted on by the Parliament. It then goes to the Senate, which is a group of retired buffoons who are appointed for life. There is a debate in Canada now as to whether the Senate is just a big waste of money, and a way for partisans of one party or another to 'kill' legislation that was passed by the Parliament.
Laws can also be struck down by the Supreme Court of Canada.
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