Originally Posted by
mh1993
1) If I use my Country-A passport (does not have any visa/PR card/permit for staying in Canada) for departure check-in, would AC staffs ask me for a valid residency permit in Canada (e.g. passport)?
Yes.
Originally Posted by
mh1993
2) If I use my Canadian passport for checking in, or if AC staffs demand that I show my Canadian passport, would they use that Canadian passport info for my API and send the API to Country-A border immigration?
It is impossible to tell unless you define Country-A.
Bottom line - when multiple citizenships are involved in a situation when dual citizenship is not possible, it is always the safest bet to go somewhere else and take the flight. For example - I always tell my friend flying from Hong Kong instead of Guangzhou because of the exact same issue.