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?
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:
If you have a base attack bonus of +1 or higher, you may draw a weapon as a free action combined with a regular move.
Note that this "move-and-equip"-rule also applies to readying a shield:
If you have a base attack bonus of +1 or higher, you can ready or drop a shield as a free action combined with a regular move.
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:
- Sheathe 2H weapon (move action)
- Move, while donning the quickdraw shield (2nd move action & free action)
- Bring 1H weapon from my gloves (free action)
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.
- Stow 1H with the gloves (free action)
- Move, while stowing the quickdraw shield and drawing 2H weapon (move action, swift action & free action)
Best Answer
In the scenario described, it seems neither PC nor foe is aware of the other before the PC walked into the room, so neither gets a surprise round.
Further, if, upon walking into a room, an armed or unarmed PC discovers a foe, both the PC and the foe should roll initiative normally. This is a battle—it may start and end as one of wits or will, but there's a foe right there with a weapon, so a battle's begun, and "[a]t the start of a battle, each combatant makes an initiative check." (However, the GM may keep secret the foe's initiative result so the PC doesn't know if he acts before or after the foe.)
Pretty much the only thing the feat Quick Draw allows a creature to do is, on its turn, draw a weapon as a free action rather than a move action; the feat does not grant super speed nor heightened situational awareness. A creature possessing the feat Quick Draw has no better chance of gaining a surprise round than a creature without it; the surprise round is for creatures aware of their foes when their foes aren't aware of them.
Also, the game lets the GM decide whether or not the ally that the foe holds at knifepoint is completely at the foe's mercy, hence helpless and at risk of a coup de grace. That is, while one GM may rule that the ally in this scenario is helpless, another GM may rule the ally is not helpless (maybe the ally is still struggling with the foe or the ally has some trick up his sleeve?). And if the GM says that ally is not helpless then the foe's coup de grace attempt is impossible, and the foe makes his attack against the PC's ally normally.