I can't say for sure that this will help, but this is a little info about Carthage based on what I remember off hand.
Carthage was founded by Phoenicians (who live in Lebanon, in case anyone doesn't know. I've actually heard a lot of people who think they were Greek). Anyway, Phoenicia was made up of independant city states, similar to Greece. Both Greece and Phoenicia would set up colonies in distant lands. However, unlike Greece, Phoenicia's colonies weren't really perminant settlements. They were more like "trading posts" where sailors could rest their ship and trade with the natives. The major exception to this was Carthage (Phoenician for "New City"), which was intended as a perminant city, in order to compete against Greek colonies. After the Assyrian empire took over Tyre and Sidon, Carthage became the most important Phoenician city, and assumed the responisbility for their protection.
Carthage also took an active role in securing distant lands for profit. The most important of these was Sicily, which was very important for its wheat production (and was the bread basket of the Roman empire from the time they conquered it, to the time they conquered Egypt). The Greeks had a foot-hold to the east with the rich city of Syracuse. Lots of wars were fought against Syracuse, but Carthage never managed to conquer it completely (the Carthaginian Sacred Band, which was a bunch of rich guys in expensive armor, gained their reputation in these wars).
After this point, Carthage decided to use its vast wealth to hire mercenaries, and the city of Carthage (which had a low population by comparison to Greek or Roman cities) was spared the requirement of military service (aside from generals). Note, other Phoenician cities were not exempt, and the natives nearby all cities were definately required to give military service. However, to pay for its mercenaries, Carthage was very ruthless to its territories, requiring them to pay huge sums of tribute, and denying them the ability to build walls (incase they rebelled). The only other city, Uttica (nearby Carthage, meaning "Old City"), was given nearly equal status, and exempt from most of these responsibilities and allowed to have walls.
Unlike Rome, Carthage's generals were not the leading politicians (The Consuls of Rome were also the head generals). While this probably helped to discourage the Sullas and Ceasars of Rome, it also left some generals without government support. Hannibal's campaigns in Italy desperately needed more men. Carthage felt its resources were better spent in securing Spain and reconquering Sicily, and didn't send many troops.
Carthage also inherited the navy of Phoenicia, which is really what sets them above. Phoenicia had a long meritime tradition as sea traders (they invented the Bireme). Carthage's navy was almost like the Brittish navy in 1800. It was used to project its power. At the height of the Athenian Empire, its navy was not as large as Carthage's.
The units I would suggest are:
Spearman: Libyan Mercenary (Numidian Mercenary). The unit Firaxis already has. The bulk of Carthage's heavy infantry was from Africa. Another possibility would be some kind of Carthaginian citizen Spearmen (which I believe existed as late as the first Punic War) or some other Punic Spearman from one of Carthage's colonies.
Chariot: I assume Phoenicians used Chariots, and Carthage might have used them when they first arrived. I think the default Chariot would be fine.
Archer: Balearic Slinger. The elite slinger and skirmisher of the ancient age was from the balearic isles and were very effective when facing off against Roman Velites
Horseman: Numidian Cavalry
Ancient Cavalry: Peonic Horseman/Punic Cavalry. Liby-Phoenician Cavalry from Carthage's colonies. Another possibility is a Sacred Band Horseman
Swordsman: Iberian Swordsman armed with either a Gladius Hispanica or a Falcata. Another possibility would be the Galic Swordsman
Knight: War Elephant (they're what Carthage is famous for).
I'll see if I can think of techs, I hope that might give you some kind of idea, though.
ARGH! I wish I could make it so that you could have two civ-specific technologies be a prerequisite for a mutual technology rather than excluding everybody from it... it'd make this so much better.
Yes, it's going to be a challenge. I've already done a lot of the research though. But some civs have been tougher than others.
