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?
Quick Draw.
The benefits don't mention putting away anything. I have always used this as a draw only feat, as I believe it's been that way pretty much since 3.0. Now, as you mentioned, that doesn't preclude you from using a free/swift action to drop your weapon, and quick draw a different one. But by default, if you want to put your weapon away, you must take a move action to do so.
Best Answer
It means you can don the shield as a swift action, while you are using a move action to move. If you do not move, drawing a quickdraw shield is still a move action.
The phrasing follows the equivalent part about drawing a weapon:
Note that this "move-and-equip"-rule also applies to readying a shield:
This makes quickdraw shields seem incredibly useless.
However, even though it is not explicitly stated, I believe these two things are mutually exclusive, i.e. you draw either a (single) weapon, or ready a (single) shield as part of a single move action. This is backed up by the part about Two-Weapon Fighting allowing to draw two weapons at once, as a shield can also be (used as) a weapon.
The quickdraw shield then has the advantage of enabling you to move whilst drawing a weapon and strapping on your shield, at the expense of your swift action.
As to your example, a quickdraw shield does not help you in the first part, since you are not drawing a weapon as part of a move action. In any case: you can't attack in that round, since you will need to move (using your converted standard action, as you have already used your move action) in order to draw the shield as a free (or swift) action. In your notation:
The other way around, the quickdraw shield DOES help. You can put away your shield as a swift action AND draw your 2H-weapon as a free action during the same move, which leaves you with a standard action to clobber your foe.