There are 2 methods to program the FX2LP.
a) Download from USB - The code can be loaded onto FX2LP from the USB interface itself.
b) Boot from EEPROM- Once code development is complete, the binaries can be loaded onto EEPROM and then system can boot from the EEPROM.
The document below has the details:
I am not sure which is the programmer you are referring to on ebay, pass the link and i can do a quick check.
If I have a circuit with a fx2lp and eeprom working. I can copy the information from eeprom and write in a new eeprom, fx2lp works as the original circuit?