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's Possible, But Tricky
You've correctly identified the difficulty of "sword and board" characters casting Shield. Shield is a great example of a spell that justifies the distinction between somatic and material components: it's a spell that is easy for non-shield wielding squishy spellcasters to cast, but difficult for the heavily armored heavy hitters (even those that can use a shield or weapon as a spell focus).
You could try dropping your sword at the end of every turn (this wouldn't cost any kind of action, as JC has clarified in tweets) and then be able to cast Shield as a reaction, and pick up your sword at the start of your turn. But this risks the possibility that an enemy could pick up your sword during their turn (either attacking you with it, or running away with it, or throwing it off a cliff etc.). That'd also stop you from making Opportunity Attacks with your sword (which also take your Reaction, but might come up on different turns than the need to use Shield).
The safest option is to get the Warcaster Feat, which would also give you advantage on Concentration Saving Throws.
Best Answer
Yes
Once initiative has been decided, the Round is made up of each creature's turns. Your turn is generally comprised of Actions and Bonus Actions, but a specific turn can include a reaction. You can only take Actions and Bonus Actions on your turn within a round. Your Reaction can be taken on your turn or on someone else's within the round.
Actions
On your turn, you are able to Move and take an Action. Examples of actions can be found in the PHB (192-3) and include Attack, Cast a Spell, Dash, Disengage, Dodge, Help, Hide, Search, and Use an Object.
There is one Action you can take that will use your reaction if you choose to use it after your identified triggering event as well: The Ready Action. Should you take this, you have an opportunity to use your Reaction to take an Action based off of a triggering event. Rules for this can be found in the PHB on page 193.
Bonus Actions
If you have an ability, spell, or some other means where a Bonus Action is available to you, you may use one on your turn.
Reactions
These are typical in response to a triggering event. Either one that you decide, As with the Ready Action or one that is used as defined by another class ability or spell. An example of this is Counterspell (PHB, 228)
As some reactions can be taken on or outside your specific turn, such as Ready and Counterspell, others can be taken later or earlier in the round, such as Counterspell or Shield - a character is able to use Actions, Bonus Actions, and Reactions during the Initiative Round.
Of course, the standard rules regarding the number of Actions or Reactions still remain and must be followed. While there may be more than one Action available to a player, there is never more than one Bonus Action or Reaction available for a player in a round.
Your examples
Example 1 - No, Tim would not be able to do this. He has spent his Action casting Fire Bolt. He can't prepare Ready Action because he no longer has an Action to spend.
Example 2 - No, Tim is still limited by his use of his Action casting Fire Bolt. If he had another Action available (e.g. Fighter's Action Surge), he might be able to set up that reaction, but it takes an Action in order to have a Reaction. There are numerous other roadblocks to why this example fails (Readied actions require completion of trigger - so a defensive buff would be 'too late' and that you can only ready a spell with a 1 action cast time.)
My Example of an Action and Reaction on same turn
Tim casts fireball at a group of enemies using his Action. It is now the turn for the enemy Wizard, who follows up with his own Fireball. In response, Tim uses his Reaction to counterspell it.