[RPG] way to sheathe a weapon faster than a move action

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?

A bloodrager that's wearing a ring of vengeful blood magic can 3/day while in a bloodrage cast a spell as an attack of opportunity. The spell must have a casting time of 1 standard action or less and either target or include in its area the creature that provoked an attack of opportunity from the bloodrager. The ring has a price of 5,400 gp.

A generous GM may allow any creature to use the ring if that creature first succeeds on a Use Magic Device skill check made to emulate the level 1 bloodrager class feature bloodrage (DC 20) to convince the ring that the user is in a bloodrage—sort of like how Lidda convinces the magic chalice that she activated it using a turn undead attempt she didn't have, an example in the Player's Handbook (2003) description of the skill Use Magic Device (86). (This example is absent from the Dungeons & Dragons 3.5 SRD, therefore it's also absent from Pathfinder.) However, this is far from a sure thing, and the GM should be consulted about this process before the nonbloodrager PC buys a ring of vengeful blood magic.

Other than this magic item, I think a player's left asking the GM if 3.5 feats can be adapted to Pathfinder. For example, the metapsionic feat Opportunity Power (Expanded Psionics Handbook 48-9) could serve as a template for a similar Pathfinder metamagic feat, as could the epic feat Spell Opportunity (Epic Level Handbook 67).

Note: Casting spells is already the game's most powerful ability, and that ability is made even more powerful if the caster can cast spells A) when it's not the caster's turn, and B) without the caster needing to expend her normal actions. A GM would be wise to strictly limit any adapted or homebrew ability of a caster to cast spells as an attack of opportunity.