Just so I make sure I am correct in my thinking
When you use the CSP card, you get points for the purchases of a travel ticket and you also get get Frequent flyer miles from the particular airline you are flying.
You can then transfer these point from your UR account to your United account and add it to your FF points thus earning you a ticket.
Is this all correct?
Yes, that is correct. So if you're flying United, for example, you would get the 2 UR points per dollar in addition to the United miles earned for the flight. If you go through the UR mall, and you link out to Travelocity, you can get an additional 2x UR points per dollar for a total of 4 UR points per dollar.
Also, you don't have to pay with a Chase card when you go through the UR mall, though it may be easier to get your points if something goes wrong and you did pay with a Chase card. If you wanted to, you could go through the Chase UR mall, link out to Travelocity, book the ticket and earn 2x UR points, but pay with another card like the Amex PRG and earn 3x MR points. Depends on whether you value 4 UR points more than 2 UR points and 3 MR points.