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?
I think you're bypassing the "don" and "doff" rule for shields.
I agree that there's daylight to be seen between carrying and "wielding" a shield. But I'll pose you this question: if you can have a shield and switch states from "wearing" to "wielding" it freely, then what is donning or doffing?
As for the comparison to equipping a weapon, I don't think it's valid. No weapon classes are described as requiring an equip-time; all armor are described as needing time to make usable. (I'm ignoring the little bit of time-constraint provided by "loading".)
My read of your description above is that you've house-ruled the donning and doffing of a shield to cost zero in the action economy. If that works for your group, that's great. But I think it is a departure from PHB rules.
Best Answer
The answer to “what is the reasonable number of free-action attacks” is simply zero, because one should not get free-action attacks. Throwing Shield should never have been printed saying that you could.
Pathfinder has a standard way of having things give “extra” attacks, and that is through “bonus attacks.” And there are several things about bonus attacks that are worth bearing in mind here:
Bonus attacks represent attacking faster somehow. You are able to simply move faster than someone else (e.g. haste), or you are able to make simultaneous attacks (swinging a weapon with each arm at the same time, say).
Bonus attacks are good. They are considered valuable class features. People go out of their way to get them. That isn’t to say that every bonus attack on offer is worth it—levels and feats are very valuable after all—but it’s not because bonus attacks aren’t good that people quibble over the costs.
Bonus attacks are starkly limited. You get one from most sources that offer them. Getting more tends to require more investment—more levels, more feats, and so on. Some sources even block you from getting more from certain other sources—consider the haste spell and a speed weapon, for example.
Bonus attacks come at a cost. A −2 penalty on all your attacks for the turn is a common one (e.g. flurry of blows, two-weapon fighting), but a spell slot is another possibility (e.g. haste), and there could be other options. A speed weapon doesn’t have a cost to use beyond being a speed weapon, but as a +3 property it’s very pricey.
Bonus attacks can only be used in a full-attack. That means a full-round action, making it very difficult to move into position if you aren’t already, and it also makes bonus attacks incompatible with various other options, e.g. Manyshot or Vital Strike.
If you have any ability to attack as a free action, the last bullet point doesn’t come into play. That means we expect the feature to have very significant limits, and/or considerable cost to use, or maybe, be available only at very high level (and still have some limits and/or cost to use).
The throwing shield is available at low level for little gold, costs nothing to use, and offers no limitations. As such, it should never, ever be offering free-action attacks. And in fact, nothing about the throwing shield makes it any faster to attack with than any other weapon you might have in your hand—so it shouldn’t even be offering a regular bonus attack either. Nothing about the description of the throwing shield suggests it should be faster than another weapon—just faster than another shield that you would have to unstrap.
And on this, Paizo’s FAQ agrees... sort of.
I say “sort of,” because Paizo attempts to worm their way out of responsibility for the error, by claiming that the text “could” mean that you get free-action attacks, and so the proposed errata would “disambiguate” things. This is an inaccurate description of the situation. The text not only “could” mean that, it quite simply does mean that—there is no version of the English language where “throw it as a free action” means “in order to throw it as a free action.” That’s not how the language works, and it is disingenuous of Paizo to claim that it does. There is no possible ambiguity here. So make no mistake, the text describing the throwing shield is simply an error. The proposed errata would be a correction, not a disambiguation. Unfortunately, this is a behavior that Paizo engages in often, and it is to the detriment of their audience because it makes us all a little less clear on what the rules are, and that the words they use mean what they say. Mistakes happen, it’s understandable and unavoidable, but they have to be clearly recognized, admitted, and corrected. This kind of weaseling out of that just makes things harder for us to play the game.
In this case, I offer that explanation of the situation because I want it to be exceptionally clear to the reader that you are not reading this incorrectly. Rather, instead, this is a situation where Paizo has written it incorrectly.