I'll start by covering the second question, since it is the simpler one to answer. In D&D 3.5 there is no concept of "targeting" or facing in the sense that you are describing. If an enemy is in combat with PC 1 and PC 2 it can be assumed that the enemy is putting forth equal effort towards fighting each. As for taking penalties to melee attacks, no you do not. In fact, based on the positions of the PCs they may get a bonus from flanking.

Now on the topic of your first question, attacks of opportunity are most commonly caused by one of two things, either moving out of a threatened space or performing a distracting act while in a threatened square. Let me define these, as stated on the SRD

Threatened Squares: You threaten all squares into which you can make a melee attack, even when it is not your action. Generally, that means everything in all squares adjacent to your space (including diagonally). An enemy that takes certain actions while in a threatened square provokes an attack of opportunity from you. If you’re unarmed, you don’t normally threaten any squares and thus can’t make attacks of opportunity.

Moving: Moving out of a threatened square usually provokes an attack of opportunity from the threatening opponent. There are two common methods of avoiding such an attack—the 5-foot step and the withdraw action.

Performing a Distracting Act Some actions, when performed in a threatened square, provoke attacks of opportunity as you divert your attention from the battle. Actions in Combat notes many of the actions that provoke attacks of opportunity.

so in the situation that you described above, being a PC moved away from an enemy, that would provoke an attack of opportunity assuming the following:

1. The PC started in a square the enemy threatened
2. The enemy had a weapon to attack with, which can include a natural weapon
3. The PC didn't do something to avoid the attack, such as a 5-foot-step, tumble check, or full withdraw action.

As for the attack being defined as free, that does not mean it automatically hits. The attack must still be rolled, applying all applicable modifiers, and any character/monster may only make a single attack of opportunity per turn. So free really means outside of the typical action economy of one standard and one move action per turn.

I hope I answered everything clearly for you, feel free to comment if you want me to explain anything in more detail. In addition I would highly recommend reading the SRD, which I linked above, to learn more about Attacks of Opportunity.

Hypothetical 1

The timing of a provocation

Casting provokes an attack of opportunity from someone who threatens you when you start casting. Note:

Concentration

Injury

The interrupting event strikes during spellcasting if it comes between when you start and when you complete a spell (for a spell with a casting time of 1 full round or more) or if it comes in response to your casting the spell (such as an attack of opportunity provoked by the spell or a contingent attack, such as a readied action).

There are two options here: either an attack between the start of the spell and its completion, or an attack of opportunity provoked (or readied attack triggered) by the action of casting the spell. That is, the full-round action you take on your turn to begin casting the spell.

So in the case of 1-round-or-longer-casting-time spells, people who move close enough to threaten do not get to take an attack of opportunity just for being there (though obviously they can attack using their turn’s usual attacks). They missed the provoking action that would have allowed them to do so.

Therefore, in this hypothetical, barring something like Spring Attack, he can’t attack while moving, and thus cannot attack the sorcerer. Attacks of opportunity never come into play.

Multiple attacks of opportunity with Combat Reflexes

Moreover, you cannot take multiple attacks of opportunity for a single provoking action, even with Combat Reflexes.

Combat Reflexes and Additional Attacks of Opportunity

[Combat Reflexes] does not let you make more than one attack for a given opportunity, but if the same opponent provokes two attacks of opportunity from you, you could make two separate attacks of opportunity (since each one represents a different opportunity). Moving out of more than one square threatened by the same opponent in the same round doesn’t count as more than one opportunity for that opponent.

Any given action can provoke at most once. This is most relevant when moving about in someone’s threatened area: they get one attack of opportunity, not one for every threatened square you leave for that same action.

Hypothetical 2

No Spring Attack, no threatened area, no attack of opportunity

Without Spring Attack or similar, this is clear:

Threatened Squares

You threaten all squares into which you can make a melee attack,

Without Spring Attack, the monk cannot attack while moving, and therefore threatens nothing. Since he cannot threaten, he cannot take any attacks of opportunity.

Spring Attack and a threatened area: still need a provocation

Now, the case of Spring Attack (or similar). The monk does threaten, and could take an attack of opportunity if one were provoked. Is one?

Readying an Action

If the triggered action is part of another character’s activities, you interrupt the other character. Assuming he is still capable of doing so, he continues his actions once you complete your readied action.

The charger does not act during the monk’s readied action. As a result, he does not move, and since he does not move, he does not leave any square the monk threatens. Thus, there is no provoking action that the monk can take advantage of.

Provoking the charger

As to whether the charger could have taken an attack of opportunity had the monk not used Tumble to get past him, yes, he could have, because he gets to layer yet another exception here:

Making an Attack of Opportunity

An attack of opportunity “interrupts” the normal flow of actions in the round. If an attack of opportunity is provoked, immediately resolve the attack of opportunity, then continue with the next character’s turn (or complete the current turn, if the attack of opportunity was provoked in the midst of a character’s turn).

So even though the charger does not generally get to act while the monk resolves his readied action (the charger’s charge is put on pause until the monk’s readied action resolves), the charger gets to interrupt the monk’s interrupt, putting his move on pause, and take the attack of opportunity. After the attack of opportunity is resolved, the turn continues, that is, the charger goes back to pause and the monk resolves his move action, and then the charger’s turn continues.

Bonus on attacks of opportunity during a charge

Finally, one last thing to note:

Attacking on a charge

After moving, you may make a single melee attack. You get a +2 bonus on the attack roll and take a −2 penalty to your AC until the start of your next turn.

I would argue that the charger’s attack of opportunity, assuming the monk had not used Tumble, would benefit from this +2. Since the charger has moved, the “after moving” clause is satisfied.

Title Question

In D&D 3.5, can a character take Attacks of Opportunity while moving without stopping?

In the hypotheticals proposed, no. In general, though? To say “no” conclusively, we would need a rule that explicitly says as much; no such rule exists. To say “yes” conclusively, we would have to demonstrate a situation in which an attack of opportunity happens for a character currently in motion.

Such a situation:

• A spellcaster readies an action to cast a spell when someone enters a square within 10 ft. of himself.

• An enemy of the spellcaster has the Spring Attack feat, and also has a reach that includes squares 10 ft. from her own position (e.g. Small or Medium with a reach weapon, Large without a reach weapon, etc.). She moves, one way or the other, and her movement takes her into a square 10 ft. from the spellcaster.

• The spellcaster’s readied action is triggered, and he takes it. The spring-attacker’s movement is paused so the spellcaster can resolve the readied action.

• The spellcaster decides, for whatever reason, not to cast defensively (maybe he does not realize that the spring-attacker has 10-ft. reach, or maybe she has the Mage Slayer feat). Thus, when he casts his spell with his readied action, he is performing an action that provokes.

• The spring-attacker, whose movement is on pause while she is in a square 10 ft. from the spellcaster, has a threatened area that includes the spellcaster. She is therefore entitled to take an attack of opportunity, and does so.

• The attack of opportunity is resolved; an attack roll is made, and if it is high enough, damage is rolled. If damage is dealt, the spellcaster must make a Concentration check to keep the spell. Regardless, at this point the attack of opportunity has been resolved.

• After the attack is complete, the readied action continues. This involves either casting the spell, or simply ending the “pause” if the spellcasting has been disrupted. Either way, the readied action has been resolved.

• The spring-attacker’s original movement is unpaused. She now gets to complete her movement.