I can see how the mandipular formation was effective against a horde of barbarians, but I'm having difficulty understanding how it would work so efficiently against a phalanx. Was it a merely, and across the board, a flanking matter? It seems, to me, that a phalanx of shields and arranged pikes, already designed to deflect arrows, would be effective against thrown pila, and that a long wall of aspides and sarisa/spear would also be effective against a scutum and gladius? I know that lack of flank cavalry and the maneuverability of the mandible was crucial to the defeat of Macedon, but what about other Phalanx armies, such as the Greek states, Carthage, Seleucids, Pontus, et al? I know that's a pretty wide range of time, but I'd appreciate any responses!
Your question is somewhat flawed in that it makes the assumption that the Rome rise to power in the Mediterranean basin was based on the superiority of the Roman fighting style. If we take a case by case looks through several decisive engagements, the picture of how Rome maintained its superiority of arms.
Hannibal and Carthage didn't employ very many spearmen in the 2nd Punic War, and was beaten by a combination of delaying tactics, and disruption of supply lines. This wasn't a maniple vs. phalanx contest, but you raised it so I will address.
The Battle of Cynoscephalae fought against Macedon, was lost when part of the Macedonian Army fell into disorder, and the romans pounced.
Magnesia fought against the Seleucids Rome had a huge strategic advantage by virtual of its limit political aims, and low perceived threat. They were able to garner a large amount of local support in Anatolia. Here it was an allied calvary command that successfully flanked the Secludes.
Lastly there is Pydna which is often cited as proof of the maniple superiority over the phalanx. And from many accounts it does highlight the weakness of phalanx over rough terrain. However, the Romans did manage to get the Macedonians to attack up hill, in a situation where their calvary would be useless, and their army would be at a disadvantage.
In all of these instances, it was the Generalship and overall leadership of the Romans that proved decisive. It was the pre-battle setup that maximized the effectiveness of the Roman Army composition over that of their opponent, and the ability of the Roman commanders to capitalize openings in battle as they appeared.
Nor was good generalship the only advantage that Rome benefitted from at this time, but that is for a different post.
The Roman Legions are known for being very versatile when led by a cunning commander. Compared to Hellenic armies, who were far less versatile, victory was a matter of superior strategy by the Romans.
A good example on how to Romans would combat a Hellenic phalanx would be the Battle of Pydna, in which the Romans (led by Paulus) attacked the Macedonians (led by Perseus). The Macedonians formed a massive phalanx outside of the town of Pydna. The Romans initially attempted a frontal assault, but were forced to retreat. Paulus then moved his troops into the rougher terrain surrounding Pydna. When the Macedonians advanced, the phalanx lost it cohesion. Gaps began to form between the Macedonian line, and the Romans sent their lightest troops (the Hasati) into the divides. This is where we see the Romans advantage over the Hellenic army model. The Romans shined in close range combat, whereas the Hellenic hoplites and pikemen relied on engaging enemies with spears. Since a pike or spear is awkward, if not impossible, to use in up close combat, the hoplites used daggers or shortswords that were less effective than the Roman gladii. In short, when fighting Hellenic hoplites and pikemen, the Romans would exploit their two greatest weaknesses. Rough terrain and melee combat. Also, velites would come in hand as well.
Now, concerning elephants, which were used by many of the diadochi as shock troops. Elephant charges were meant to shatter an enemy line and demoralize their troops (it's hard to keep your composure when an elephant just stepped on your buddy). Like phalanxes, Roman maniples adapted to enemy elephant cavalry. At the climax of the Second Punic War, at the Battle of Zama, Hannibal had the advantage with his war elephants. Scipio Africanus countered this by having his frontline dodge the Carthaginian elephants (as they charged in a straight line, dodging involved forming corridors for the elephant to run through). Once the elephants were trapped behind enemy lines, the Romans used their pila to bring the beast down from a safe distance.
I'd also like to make a note of the Carthaginians in comparison to the other Hellenic powers you mentioned. While the Carthaginians are normally grouped in with the Hellenics, their armies were vastly different. While the Hellenic armies were made up of citizens, Carthaginian armies were composed of Celtic and Libyan mercenaries. Compared to the armies of Greece, the Seleucid Empire and Ptolemaic Egypt, the armies of the Republic of Carthage were vastly different than their eastern counterparts.
I notice you're using "mandible"/"mandipular" consistently (which is nice; if you're going to spell something wrong, at least do it consistently :-). The correct terms are maniple and manipular, without the 'd'. Just wanted to make sure you're aware of that.
Also, in addition to the answers already given here, check out my reply to the question "How would a Roman legion defeat a Hellenic phalanx?" from a few months ago.