I'm assuming by Non-EU 3rd country certificate, you're referring to the Regulation (EC) No. 998/2003 form that says on the top "Veterinary Certificate for Domestic dogs, cats, and ferrets entering the European Community and Norway for Non-Commercial Movements?" If this is the form you are asking about, it doesn't matter how many pages it prints on. Mine is printed on 3 pages.
Anyway, also that form you can't have the section OFFICIAL VET/OR VET AUTHORIZED BY THE COMPETENT AUTHORITY, I'm not sure it matters when it's signed, but it needs to be sent to the USDA for stamps and a signature, so I just waited to have it signed until 10 days prior to my trip. Parts I-V can be filled out before hand because it's all the identifying information. The OFFICIAL VET/OR VET AUTHORIZED BY THE COMPETENT AUTHORITY section is signed by your local vet and the next section Endorsement by the Competent Authority... is signed by the USDA of your state. And the final two sections (Tick and Echinococcus treatments) are filled in later 24 hours before you depart.
Now the Aphis form 7001 definitely cannot be signed until 10 days prior either because it's only valid for 10 days when you are travelling internationally. That form you also send to your state's USDA and they sign and stamp it.
Hope my explanations weren't too confusing