No problem.I do this all the time when going to the US. I enter the US on my US passport and enter Canada on my Canadian passport. There is no exit control for either country anyway.
Yes, it's fine. If you are traveling to the ciountry where you are also a citizen you should enter that country on your passport for there. When traveling back to Canada you must enter on your Canadian passport.
A lot of airlines will ask you to prove that you can travel to your destination when you are checking in so they may ask you to present the passport then. For example I am PR and an Irish citizen. When traveling back to Canada I have to show them my PR card at check in or they will not let me board my flight. If I had a work permit I'd have to show that.