Are you sure that you are eligible for the PEQ? The PEQ is for graduates holding a diploma from a Québec institution, so while your diploma from BC would get you major points in a normal CSQ application, it does not make you eligible for the PEQ. See also: immigration-quebec.gouv.qc.ca/fr/immigrer-installer/etudiants/demeurer-quebec/demande-csq/etudiants-peq/diplome-admissible.html
Edit: Unless, of course, you are taking the other PEQ track, in which case you need to meet the specialized work experience requirement: immigration-quebec.gouv.qc.ca/fr/immigrer-installer/travailleurs-temporaires/demeurer-quebec/demande-csq/travailleurs-peq/emplois-admissibles.html
To answer your two questions though:
1- The Immigration-Québec process yields the CSQ, which allows you to proceed to the federal application process. Once you receive the CSQ, you will have to manually start your application with the CIC. But keep in mind, since Quebec has selected you already, you are only subject to security and admission controls at the federal level - there is no interview or selection criteria requirements at the federal level for those holding a valid CSQ.
2- The CSQ application in the PEQ program is one time. A normal CSQ application can have stages, including an interview process.