I'm not 100% sure what you're asking. There is no requirement to be employed in order to maintain permanent resident status. So it doesn't matter which of you gets the job. A job offer can help with points before you become a permanent resident, though.
You can, in theory, apply for different streams at the same time. I think the one you don't apply for would just be abandoned if you got accepted for the other. But if you have the invitation to apply, and you qualify, why apply for two?