Is an attack of opportunity triggered by maintaining a grapple?

Question

Scenario: A character A is maintaining a grapple of another character B (who is currently stunned). The grapple has already been initiated the round before. A third character C is in melee distance of A.

During A's turn, A takes either the "Attack Your Opponent" or "Damage Your Opponent" option from the actions that can be taken while in a grapple.

Is C allowed to make a attack of opportunity on A?

I haven't found any clues in the description of grappling. Another source I found is table 8-2, “Combat Actions”, in the PHB that states for a Grapple either:

1. (1) Regardless of the action, if you move out of a threatened square, you usually provoke an attack of opportunity. This column indicates whether the action itself, not moving, provokes an attack of opportunity.

2. (7) These attack forms substitute for a melee attack, not an action. As melee attacks, they can be used once in an attack or charge action, one or more times in a full attack action, or even as an attack of opportunity.

It's seems to me that the bolded parts clash, the first granting a generic AoO and the second denying a generic AoO.

Also, others questions about grappling in almost same situation:

1. C attacks A with success: does the grapple on B end? [no, in my understanding, grapple is independent from damage taken by winning grappler]
2. C joins the grapple A [automatically]: how the grapple on B can end?
3. Another character D performs a winning Bull Rush on A. Does the grapple on B end? [In my understanding, D is bull-rushing both of them]

Answer 1

When Table 8–2: Actions in Combat (Player's Handbook 141) indicates that a grapple provokes attacks of opportunity, it's indicating that making a grapple attempt to start a grapple typically provokes attacks of opportunity. A creature that wants to grapple actually obeys the longer and more detailed rules for the special attack Grapple (PH 155-7), the table providing only a summary. Other actions listed in the section If You're Grappling provoke attacks of opportunity only if those actions say they do, not—if following only the table—for every action taken during a grapple.

• If Creature C makes a successful attack against Creature A, unless Creature A is slain by the attack, Creature A and Creature B continue grappling normally.
• The Joining a Grapple rules would have Creature C participate in the grapple normally. Creature C's presence adds a layer of complication to the grapple, but doesn't change fundamentally how grapple works.
• The special attack bull rush affects only one creature: "When you make a bull rush, you attempt to push an opponent straight back instead of damaging him" (PH 154 and emphasis mine). If Creature C were to succeed on a bull rush against either Creature A or Creature B, their grapple likely ends.

Answer 2

Yes

The rules for making an attack in a grapple are as follows:

Attack Your Opponent

You can make an attack with an unarmed strike, natural weapon, or light weapon against another character you are grappling. You take a -4 penalty on such attacks.

You can’t attack with two weapons while grappling, even if both are light weapons.

Damage Your Opponent

While grappling, you can deal damage to your opponent equivalent to an unarmed strike. Make an opposed grapple check in place of an attack. If you win, you deal nonlethal damage as normal for your unarmed strike (1d3 points for Medium attackers or 1d2 points for Small attackers, plus Strength modifiers). If you want to deal lethal damage, you take a -4 penalty on your grapple check.

Exception: Monks deal more damage on an unarmed strike than other characters, and the damage is lethal. However, they can choose to deal their damage as nonlethal damage when grappling without taking the usual -4 penalty for changing lethal damage to nonlethal damage.

There is no mention of either of these actions triggering an opportunity attack - so this is treated as a normal attack, except for the -4 modifier and the restrictions on which weapons can be used. However, Table: Miscellaneous Actions in the Actions in Combat section says that taking the special Grapple action itself provokes an attack of opportunity; so C can make an attack of opportunity. A will also lose their Dexterity bonus to AC against C's attack. C can choose to have their attack be joining the grapple, if they want.

(Thanks to @Ols for pointing out that B being grappled prevents them from making attacks of opportunity.)

To answer your other points briefly:

1. Taking damage doesn't affect a grapple unless it reduces you to 0 or fewer hit points.
2. If C joins the grapple, they become part of the grapple, but they are on B's side, so B still needs to do the same thing to escape the grapple. B needs to (as their action) choose to "Escape from Grapple" and then score higher on their grapple check than any opponent in the grapple that tries to stop them. (If E was helping A grapple B, B would have to beat both A and E's opposed rolls to successfully escape the grapple.) If B passes this grapple check, he moves out of the grapple space into an adjacent space of their choice.
3. This case isn't explicitly covered by the rules, as far as I can tell.

(If you want more detail on these points, you should probably make them their own question.)