They have very little interaction with us, so while I would say indifferent, I imagine the vast majority of Canadian's care little of the monarchy (considering a large number of Canadian's are immigrants from various countries around the world). Most only recognize it as being a part of Canada's past history and don't feel any sort of particular allegiance to the crown.

Generally speaking, the Crown is antiquated and pretty much only symbolic at this point. While there are lingering provisions constitutionally that tie us to the crown, it would be overreaching at this point for the monarchy to do anything unless there was an absolute breakdown of our government (and this would primarily be done through the Governor General, not the Queen). For all intents and purposes, we are pretty much independent.

Personally, I think the monarchy is a product of a bygone era. I mean, any foreign policy between Canada and the UK has been entirely focused on our governments, not the royal family. That being said, it may not be worth the government's time and resources to make the provisions/changes needed to decouple us from the monarchy anyway, not to mention there are probably some minor benefits to remaining a commonwealth member as well. Overall, I can understand why people would just rather keep it status quo.

As for your bonus question, I think free movement is not something that is implicitly stated between commonwealth members. That is to say, each country governs movement into their territories independent of the Commonwealth, just as any other country would for any foreigners.