You ask a lot of questions (and I have some myself, and I'll try and answer one.
Assuming that Canada became independent in:the late nineteenth century, how would be its relations with the US? Could attract enough emigrants? Would become a stable state? Would emerge a manifest destiny to the French canadians? Would emerge a French US?
Completely depends on the PoD. If by Canada you're referring to what becomes Lower Canada, perhaps a successful rebellion of 1832 then no, I highly doubt they'll be able to expand any territory southward. You'd need an earlier PoD for more of that southern land, and you'd definitely need a
much earlier PoD then the 1800s if you want a French colony resembling anything like the United States.
Manifest Destiny would get butterflied away in that case. A similar situation or method of thought of course could come up, but it would be a different means ittl. In terms of immigrants: a Catholic state would be a huge beacon (and it was iotl) for people coming out of places like southern Germany, Ireland, Italy, France itself, etc. The United States had always been fairly hostile to Catholic immigrants. I doubt Canada/Quebec could compete with them in the long-run population-wise, and the U.S. would still probably get more Catholics in total numbers than Canada: but Canada would get a lot imo, enough to keep the country relatively populous.
The problem isn't getting population though, it's what happens when it becomes overpopulated. Iotl there was a huge exodus of French-Canadiens in the mid-1800's; partly because of discriminatory policies by the government, but mostly because jobs weren't that great in Canadaand farmland was pretty entrenched and unavailable by that point. A Canada with an earlier spike in population would get this earlier (likely), which could lead to conflict elsewhere. But it totally depends on what land Canada has.