Yes, it's the capital.
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.
It depends, Quebec is a province in Eastern Canada, whilst Quebec City is the capital of said province, but is sometimes alternatively labeled 'Quebec'.
T-shirts and jeans. Shorts in summer. Coat and boots in winter. Nobody wear "ceintures fléchées" or "beaver hat" as you may think ;)