Yes, it's the capital.
Quebec!In french it's easy to know if they are talking about the province or the city because of the word they put before "Québec" :« Je vais à Québec » I'm going to Québec (the city).« Je...
It became a city in 1663.
The capital of Quebec is Quebec City.
No. Only Old Quebec which refers to the area within the city walls plus the Petit Champlain district next to the river.