PFA,
If you are getting a reman 01 L motor don't worry because it is actually a rebadged 01 new motor. Ford sent the remaining engines to A.E.R. (Ford's rebuilder) and they just remove the stickers on the valve covers, add in a few necessary gaskets and affix a 1" x 1" metal plate to the RH side of the block that has "Remanufactured" and the 01 part number 1L3Z-6006-HARM. Mine was just replaced due to an oil consumption problem as well as a fellow board members, both were new motors and as described above.
As far as buying your core back, current core charges are $1000 and I was told by the Remanufactures rep that the warranty claim would be voided (mine was right at $4400) if they did not receive the core back. LightningJeff's deal sounds a little different as his motor was sent directly from the assembly line as it included the blower, harness, ect. In his case someone in the parts dept made a quick $100 selling what was considered scrap, not a bad deal for either party, but much different then what is taking place now.