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 conditions that prevent characters from taking actions altogether (stunned, unconscious, dying, etc). This prevents free actions, but a "No Action" power could still be used. Also note that most powers that enhance your initiative are No Action because you are using them before combat technically begins.
For example, a Deva could use the Memory of a Thousand Lifetimes power to boost a saving throw taken while stunned, but a Battlemind couldn't use Battle Resilience while stunned.
My guess at the reason this particular power is a No Action power is that you can normally only make one attack as a free action per turn. "No Action" powers don't count toward this limit.
Free actions take almost no time or effort. You can take as many free actions as you want during your or another combatant’s turn. There is an exception to that rule: A creature can take a free action to use an attack power only once per turn. Creatures don’t normally have attack powers that can be used as free actions, but some powers and other effects grant the ability to use an attack power (usually a basic attack) as a free action. For example, a character might have two different abilities that let him or her make a melee basic attack as a free action when their respective triggers occur. If both abilities are triggered on the same turn, the character can make only one of the melee basic attacks during that turn.
Centered Flurry of Blows could be argued to be an attack power, and thus use up your 1 free action attack per turn. Giving it a "No Action" ensures that it doesn't.
Best Answer
Officially, this is one of the handful of things the game says that a lone typical creature just can't do with a typical weapon…
As per this excellent answer, in official material there are ways for the typical creature to sort of take a free action to sheathe a weapon, but these are largely workarounds for an option never officially never provided beyond the gnome quickrazor (Races of Stone 154, 155) (45 gp; 1 lb.).
(I've always assumed that the inability to sheathe a weapon as a free action was to prevent overuse of the skill Iaijutsu Focus (Oriental Adventures 58-9, 81-2), which says that an "attack [against] a flat-footed opponent immediately after drawing a melee weapon… can deal [up to +9d6 (or, at epic levels, even more)] extra damage, based on the result of an Iaijutsu Focus check" (81), therefore requiring a newly unsheathed weapon to be sheathed again before the skill can be used a second time with that same weapon. Shockingly, a warrior doesn't usually want to drop her weapon after making that one attack (or, as some argue, her whole round's attacks) and use a different weapon; she'd rather find a way to sheathe and redraw the same weapon. The game knows this, and, even at the bitter end, sheathing a weapon as a free action remained one of the few officially-almost-impossible D&D 3.5 tasks.)
...But third party says, "Meh. Whatever."
With third party material available, this becomes much easier. For example, Paradigm's Arcanis Player's Guide (2004) includes this Open-Gaming-Content feat:
(The Arcanis Player's Guide also describes the feat Quick Hands (186), but the feat Quick Draw (Player's Handbook 98) is typically a better choice for meeting this feat's prerequisite.)
I am certain other third party texts provide similar options, but this is the first I happened upon.