Is the EEA Family Permit the correct way to go?
Yes.
If my son gets the Italian Passport on time for the trip, does he need the EEA Family Permit?
Of course not. The EEA family permit is only for non-EEA family members of EEA nationals. As an Italian citizen, your son does not need an EEA family permit. Focus your efforts on getting the Italian passport instead.
Do we need two applications, One for me and one for my son, or can the application include both?
You would need two applications, but your son should get his Italian passport instead of the EEA family permit.
As I understand we must apply in person (after the online process) in Buenos Aires, Argentina, because there is no office in Montevideo; is this correct?
Perhaps not. The freedom of movement directive requires member states to "grant such persons every facility to obtain the necessary visas. Such visas shall be issued free of charge as soon as possible and on the basis of an accelerated procedure." This generally means that you cannot be required to pay a "processing fee" for the services of a third-party visa processor. It may also mean that you shouldn't be required to travel to Argentina. You should try asking the nearest consulate it will accept your application directly.
If my son must apply for the permit, must he come with me to Argentina? Must he be there for the application?
If you can't get his Italian passport in time then he probably has to visit. The consulate or the visa processing contractor can give you more details.
Is our son's birth certificate proof enough of our relationship? If not, what additional documents should we provide to prove the partnership?
What are the requirements for the documents presented (translation?, any official validation?,apostille?)
Are there any additional documents we must provide with the application?
If you are unmarried, and your partnership is not registered with any civil authority, include evidence to show that you have been living together for at least two years. (I suspect that it should not be necessary to prove this because you have a child together, but I cannot find any official guidance to support that suspicion. Having a child together should at least relieve you of the burden of providing additional evidence that that your relationship is genuine.)
If you have registered your partnership, include authenticated copies of the registration documents. If you are married, include an authenticated copy of your marriage certificate.
Regarding the requirements for certification and/or authentication, and more generally for information about what documents to include, see the guidance and the documents to which it links.
