When you do a Chase PC the old card (CSP in this case) will stay active for approximately 45-60 days during the transition process. After those 45-60 days, the old card will automatically become inactive and disappear from your account. When I did exactly this (PCing my CSP Visa to a Freedom MC) the CSR told me that I could continue to use my CSP with the same rewards structure during those 45 days, but it appears that it switched over to the Freedom reward structure immediately. So when I was physically swiping my CSP Visa, it was like I was swiping my new Freedom MC when it came to rewards.

Also, I suspect the CSR was mistaken when she asked if they could move the $5k limit. It would have likely produced an error if she tried to execute that change.