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 is no definition of actions requiring "attention", so I believe it's safe to assume that sentence to be descriptive.
This means that there are two limits :
Anything not included in these categories is doable.
Regarding the specific acrobatic case:
Action: None. An Acrobatics check is made as part of another action or as a reaction to a situation.
He should be able to use acrobatics as part of his single "Move" action, note however that
In addition, you can move through a threatened square without provoking an attack of opportunity from an enemy by using Acrobatics. When moving in this way, you move at half speed. You can move at full speed by increasing the DC of the check by 10.
So he either took the penalty or moved at half speed.
In any case, since he got "past" you using his only available action, he shouldn't be farther than a single move action, and the PC should have no trouble catching up to him.
The only way for him to 'run' away, excluding magic, is for his normal movement speed to be more than twice the PC's speed.
https://paizo.com/pathfinderRPG/prd/skills/acrobatics.html
Best Answer
RAW: No
As you mentioned, the description of Nauseated clearly states that you can only take move actions, and a down-conversion of move->swift is not possible by RAW.
Take a look at the very similar Staggered condition, where free/swift/immediate are explicitly allowed, implying that needs to be the case.
(emphasis mine)
RAI: Maybe?
People in this paizo thread point out that there are some class features and spells that imply that swift actions should be possible.
The Persistence Inqusition has this ability:
The "Range: Personal" spell cleanse:
Personally, I find these arguments very weak. Neither of these abilities are specifically designed to counter Nausea, both are rather cleansing a broad variety of status effects. The spell argument is further compromised by the fact that casting spells is also explicitly forbidden while Nauseated, which definitely points to developer oversight, which are not uncommon[citation needed] with such an expansive ruleset. By extension, I guess the author of Inner Strength didn't expect the Persistence Inquisition not to work correctly.