Can a legendary monster/NPC use a “free object interaction” as part of a legendary action

Most free actions must occur on your turn, so no, the Dwarf cannot draw his shield

I quoted 3.5 in the other question, but since you're not interested in that, I'll stick with a reading of the Pathfinder PRD:

Combat Round:

In a normal round, you can perform a standard action and a move action, or you can perform a full-round action. You can also perform one swift action and one or more free actions. You can always take a move action in place of a standard action.

Free Actions:

Free actions don't take any time at all, though there may be limits to the number of free actions you can perform in a turn. Free actions rarely incur attacks of opportunity. Some common free actions are described below.

Great, free actions are quick. What's speaking?

Speak

In general, speaking is a free action that you can perform even when it isn't your turn. Speaking more than a few sentences is generally beyond the limit of a free action.

(Emphasis Mine)

So, why is there a special clause for speaking that says you can do it when it is not your turn if you can take free actions outside of your turn? This clause does not come into conflict with "You can perform one or more free actions while taking another action normally.", which is simply a clarification that "Sure you can drop your sword while attempting to climb the rope while shouting at your Dwarf companion to get his shield out"

How about immediate actions?

Much like a swift action, an immediate action consumes a very small amount of time but represents a larger expenditure of effort and energy than a free action. However, unlike a swift action, an immediate action can be performed at any time—even if it's not your turn.

There again is the special emphasis that it can be done when it is not your turn. This further implies that most actions can only be taken on your turn.

Paizo, to my knowledge, has not commented on this because it is the same as 3.5, and this has already been covered in a FAQ by the 3.5 designers. Since Pathfinder is an improvement on 3.5 and doesn't change the rules wholesale, why would they comment again on this?

Source? Pathfinder PRD combat section.

And for those interested, further discussion on the 3.5 side of things here: https://rpg.stackexchange.com/a/8896/1736

Edit

And to counter a point from your question:

This interpretation also makes sense stylistically, as a character should be able to take their usual free actions while already reacting to another character, such as drawing a shield while intercepting an attack or loading a crossbow with rapid reload while making an attack of opportunity.

What about during the surprise round? Say someone botches their perception and is about to get shot with said arrow. Totally unaware. Should he be able to whip out his shield then?

There are no specific rules vis a vis how a character starts combat. D&D5E attempts to avoid that kind of specificity when basic logic and reasoning will do:

• If the character is hanging out in a tavern, his hands are empty... or full of food. Utensils are Improvised Weapons, by the way.
• If the character is hanging from a chandelier, at least one hand very well ought to be weaponless (in so far as one does not wield a chandelier, it had best be making use of one hand).
• If the character just finished picking a lock, his hands are empty.
• If the character just finished picking his nose, at least one hand is empty.
• If the character is dungeon delving and carrying a torch or lantern, he's probably got a weapon in his other hand or has a shield strapped to his arm.
• If the character is dungeon delving and doesn't need a hand for a light source, he'd be smart to have both weapons out.

Either way, it's a matter of common sense and story description, not codified rules.