If he is prone, tripped, does he still threaten nearby squares and do I provoke an attack of opportunity by attempting to grapple?
So yes can still take an AoO, but he takes a -4 penalty on that attack. He also takes a -4 penalty on his AC.
If I succeed in grappling, can I immediately roll again to see if he's pinned or does that take another round?
That will take another round.
Do grappled and prone penalties stack to a -6 on CMD when attempting to pin him?
The combat maneuver penalty for being grappled does not count towards any grappling checks (and pin is explicitly stated to be a more severe form of grapple, so any rule that applies to grapples also applies to pins) so you only get the -4 from.
Once pinned, does it take yet another CMB to tie him up or do we just do it? What would the penalties be on an NPC prone, grappled by one character and pinned by another?
No, you only get the penalties for prone. "Tie Up" works like "pin", which works like "grapple", and the combat maneuver penalties for grapple do not apply to grapples.
One of our party has successfully tripped an attacker and I want to capture him. If I understand this, we have to grapple, pin, and tie him up.
Correct, though beating him until he surrenders is also an option, depending on the GM.
If he is prone, tripped, does he still threaten nearby squares and do I provoke an attack of opportunity by attempting to grapple?
Yes. The prone condition gives him a -4 penalty on his attack rolls, but does not stop him from threatening squares or taking attacks of opportunity.
If I succeed in grappling, can I immediately roll again to see if he's pinned or does that take another round?
This depends on your character. Assuming that you have no grappling feats, then no. You spend a standard action to grapple him, then wait until your next turn and spend another standard action to maintain the grapple. If you successfully maintain the grapple, one option you can take is to pin him. Once he is pinned, you must take another standard action (and thus, wait another round) to tie him up.
Note that characters who invest feats in grappling become much quicker at it. Greater Grapple allows you to use move actions to maintain, which would allow you to grapple as a standard, then maintain and pin as a move on the same round. Rapid Grappler allows you a third grapple check as a swift, so with enough investment, you could grapple, pin, and tie up an enemy all in one round.
If it takes another round, can a -third- ally pin him while I have him grappled?
No. Only one creature can make the checks in a grapple, though others can use the Aid Another action to grant you a +2 bonus on your checks.
Do grappled and prone penalties stack to a -6 on CMD when attempting to pin him?
Assuming that by the grappled penalty you mean the -4 to Dexterity, yes.
Once pinned, does it take yet another CMB to tie him up or do we just do it? What would the penalties be on an NPC prone, grappled by one character and pinned by another?
Yes, tying up a pinned character takes another maintain-grapple check. Prone would give a -4 penalty to AC, while pinned would give an additional -4 as well as denying the enemy's Dexterity bonus. Pinned does not stack with grappled, being the upgraded version of that condition.
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Remeber that Caragorn's target gets a bonus to his CMD versus Caragorn's grapple attempt equal to the damage he did if he hit Caragorn on the AoO.
And that his AoO doesn't, necessarily, have to be an attack for damage. He could also try to trip, disarm or sunder as an AoO, although that might provoke an AoO from Caragorn...
My advice, in a situation like that, is to, instead, take the -4 penalty to atack, which is a wash with the bonus for attacking a prone target, with a non-lethal attack. Knock him out, and he is easy to tie up after that, and smelling salts can let you wake him up without giving him any hit points back...
Or use an attack with a net, which is against touch AC, so even the -4 penalty against a prone target with a ranged attack is still fairly likely to hit. Nets apply the entangled condition, which is similar to the grappled condition...