When I flew, I had to produce the following when I checked in at BA World Cargo:
+ APHIS 7001 form, signed by me, the local vet, and the USDA vet, and embossed by the USDA vet
+ EEC 998/2003 form, signed by me, the local vet, and the USDA vet, and embossed by the USDA vet
+ Rabies vaccination certificate - the paperwork from when the vaccine was given (not sure that everyone has to produce this, as the information is contained on both the APHIS 7001 and EEC 998/2003 forms, but I was asked to show mine)
+ Original results of the FAVN-OIE Rabies Antibody test from the lab at Kansas State
All other forms were provided for me at BA World Cargo and I filled them out there when I dropped the boys off. You might want to give Continental a call to see if they require any additional paperwork, especially if it's something they want done ahead of time and faxed to the UK (I had to complete a C5 form and fax it to Gatwick before our flight).