The answer to all your questions is ‘yes’, provided your girlfriend gets her visa and successfully gets to Canada:
- Yes, you can marry each other in Canada, provided you can both prove your identities and that you are free to marry (i.e. not currently married).
- Yes, she can stay here while you sponsor her. It’s called “inland” sponsorship; it takes longer but she can stay with you.
- Yes, she can work, provided she submits a work permit application at the same time as the sponsorship application.