@Ifusaso answers correctly, but here are direct answers to all your specific questions (hopefully):
How to gain access to things that require you to be a member of the Pathfinder Society?
You aren't far off. You merely need to say "My character did that", and have the GM agree with you. Unfortunately the Character Guide isn't fully on Archive of Nethys, but if we pull this excerpt from the Advanced Player's Guide on Access Entries, we see:
Sometimes, a stat block for an uncommon rules element includes an Access entry that lists specific criteria. A character who meets the criteria listed in the Access entry, such as hailing from a particular locale or being a member of a particular organization, gains access to the rules element. For instance, a draconic bloodline sorcerer gains access to the dragon disciple archetype, even though the archetype is uncommon.
As always, the GM has the final say on who can access uncommon or rarer options, or whether specific options are allowed in the game at all. They may decide that a character's upbringing or abilities make sense for a specific rules option and grant them access even if the character doesn't automatically qualify, or they may decide that same option can't be taken by anyone at all.
While the archetype lists being a member of the Pathfinder Society as a "prerequisite" and not under Access, the same rule of thumb applies. A GM could homebrew a general feat
Does every member of the Pathfinder Society (the Golarion organization, not the IRL one) have the Pathfinder Agent archetype?
No. There is nothing that says a member (aka agent) of the Pathfinder Society must take the Pathfinder Agent dedication. Just like not everyone dedicated to the bow must take the Archer dedication (which references "those dedicated to the bow"), nor must everyone who is a student of martial one-on-one combat a Duelist (which references "students in martial academies practice with their blades to master one-on-one combat"). So we cannot assume that a feat's description text mandates that a character that meets said description must take that feat.
Why does Pathfinder Agent require you to be a member of the Pathfinder Society?
This is a confusion of prerequisite and mandate. There is no text mandating that a character, even an agent of the Pathfinders, take the Pathfinder Agent dedication. The dedication simply allows characters who meet the prerequisite to take it. To gain these skills, you need the kind of training the Pathfinder Society provides; you don't not need the training to be a member of the Pathfinder Society.
Do you need the Pathfinder Agent archetype to access things that require being a member of the Pathfinder Society?
Per my earlier statements, no, since you don't need the archetype to be a member of the organization. Feats like this (such as Fane's Fourberie) specify that to access it:
Access: Member of the Pathfinder Society.
It does not say:
Access: Pathfinder Agent Dedication
Prerequisite: Pathfinder Agent Dedication
GM can allow access to all of these things, and the Pathfinder Agent archetype isn't needed to be considered a member of the Pathfinders.