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:
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 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
Great, free actions are quick. What's speaking?
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.
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
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?
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
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?
Unfortunately, it would require your action. You can sheath a weapon or draw a weapon using your free object interaction, but not both. (Addendum: The Dual Wielder feat allows you to draw 2 weapons or sheath 2 weapons using a single object interaction. Note that this does nothing to fix your problem, it just seemed worth mentioning.)
A Rogue with the Thief archetype is absolutely free to use their Cunning Action to Use An Object on any object they choose (assuming they have the feature, obviously), and the scenario you describe is a perfectly legitimate use for Cunning Action.
However, there is another solution for all the non-Rogue players who want to switch weapons and attack in a single turn: Drop your weapon, don't sheathe it.
Dropping something requires no action at all, not even your free object interaction. So you can drop your weapon, use your free object interaction to draw a different weapon, and attack with that weapon on a single turn. And move too, if you want. Just remember to come pick up your dropped weapon later (probably at the end of the combat.)