I'm also not a huge fan of that. They should be allowed to become a warmongering powerhouse through economic strength though. (just realized you meant Quinqueremes. I like +2 sight +2 movement -> as promotion or just a bonus to the unit? If they are to be great scouting tool, maybe allow them to enter ocean at compass? Or supply so that they could survive in distant lands?)
How does the unhappiness actually work? I don't really understand that in CBP. Small unhappiness doesn't really hurt the empire all that much, from what I've seen, so I am not sure if that would be a lever which wouldn't allow Carthage to go crazy. Notice that you have to expand and expand a lot to really make use of this UA. 10 puppets (break even with Venice)? Where would you find place for that, especially at higher difficulties? Only one tile islands might be left. And it could still be balanced with pricing. Losing a ship and paying 300/400/500 gold is not a small amount. The price could rise with each consecutive city, and perhaps should rise with era. Cothon itself would cost 400 (or more) to purchase in a city and I wouldn't count on autogovernor to build it quickly (unless the building flavor would be changed). Venice has 4 trade routes with just sailing and animal husbandry, getting 2 more with each trade tech. Carthage won't get two bonus TRs by the time it gets optics because first, it'd need to research sailing, second, build two quinqueremes, third, pay for the colonies (lets say 700 gold, 300 and 400, even though at first I was thinking of starting at 200 gold cost at first), fourth, pay for the Cothon (800 gold at least). Putting a significant gold burden here is a way to balance the UA. Venice also gets a city state with all of its units, buildings, population, land and improvements. Carthage would have to start the city from scrap.
Honestly I would even consider ditching the free harbors for that (though it became sort of a trademark of Dido). I don't think depriving them of settlers is an option here because then it would mean no cities whatsoever if landlocked, and starting inland can happen. I feel the UA fits really well here, as you'd get the city of Carthage with the core part of the empire (a couple cities, which historically were in modern Tunisia and west part of Sicily) and large puppet portion (all the trade colonies along the coasts of the western Mediterranean). And you'd have a reason to expand along the coasts to boost the trade of the empire (the trade route could be easily flavored as expanding their markets by trading with the locals). Allowing purchase in puppets is just to make it playable and it still fits due to Carthage's historical mercenary army (to the point where I would consider giving them a penalty on producing land units - Carthage never really had large citizen army). And the fleet utilized for expansion - that's how they expanded historically, and how Carthage itself was founded.
Even their navy's main task was maintaining their trade superiority in the Mediterranean, so a bonus for combat inside their territory still fits.
Gah, but all that still doesn't make the terribleness of the AI any better.
Ok, now about your gold proposition:
What I don't like about slamming gold on water is that it's simply boring. And it would still be better to work any plains tiles if they're available (with 3f 1p they're superior). If we were to go the gold route I would additionally place flat production on cothon and give it a military bonus, or even incorporate something like +2 production from naval buildings into the UA. When we look at the Ducal Stable it gets gold on resources and experience boost, which is a bit less, but Poland has one of the strongest UAs in the game. The Mayans get a shrine with 2 science and +1 faith, that early it's insanely strong. Not to mention the king of UBs, the Bazaar, with additional copies of luxuries (7gpt for each luxury or 4 happiness... Or strategics) and 2 gold on oasis and oil tiles (which alone could be UB, not a great one, but it could). Arabia's UA is of course worse than Polish, but their UU and UB are amazing. I think every civ should feel this way - that they have something amazing which is unique.
Note that in GnK Carthage actually had an early production bonus because harbors gave +1 prod to sea resources, making fish, whales etc. good tiles right away (whales being salt equivalents). Production from harbors isn't gamebreaking because there is no harbor at turn 1 (+2 production would be too strong if available since turn 1, I played with a mod that added that to harbors and earlygame Carthage was insane with that production. I could just spam nearest neighbour with units thanks to that early infrastructure I got that much faster than other could. It was just king or emperor though).
Alright, other ideas... What about giving the Cothon a city connection gold modifier (useless in capital), or making it a naval bazaar (additional copies of luxes, takes away uniqueness from Arabia so probably not), larger bonus to trade route gold, more range for trade routes (kind of meh), internal trade route effectiveness modifier (interesting, but it would serve creating super cities, not trading with others)? Culture from harbors could still work, but it's not an economic boost. Cothon could also just add +25% gold in the city, but it'd be bad in cities with low income. Unless combined with the gold yield on sea tiles? Maybe add merchant specialist slot(s)? Engineers and Scientists are a lot better... Maybe make their UA improve merchant specialists as well (more gold, more production, maybe some culture)? It'd be nice if they were actually useful for a trade empire. Gain a trade route slot for each great merchant generated? Scrap the quinquereme and give them a great merchant replacement who can sail like a great Admiral? Give great merchants an ability to found trade colonies which are puppets (and then add the trade route on Cothon). Or both - merchants sail like admirals, while both admirals and merchants can found trade colonies. Making great merchants useful for Carthage might be the way to go
Another idea: UI like Feitoria, but instead of granting luxuries it'd incorporate the nearest city to the carthaginian city connection network. Could be built in foreign civilization lands (milking that huge capital with Hanging Gardens and Temple of Artemis) and would provide significant gold when worked (dependant on something like number of resources nearby, era, size of the city). Called just "Trade colony" or something like that. If buildable by great merchants/admirals, could provide additional trade routes, but then would probably be buildable only in city states territories. If you send a trade route to such a city state, gain influence/turn (like freedom policy).
Woah. From merchant specialists to unique improvements. Dat train of thought.
As for Cothons place in the tech tree, I think Commerce is available later than sailing, in one column with optics. I would put Cothons on sailing and harbors on optics probably, to maintain the order. Maybe seaports available at compass? xD
[The post was edited 3 times, each time adding new ideas. I hope the whole is still readable after inserting more ideas in places I thought are right, but well, It's before 8 AM here and I started writing before 4 AM]
[EDIT] I just noticed that the merchant slot was already present in Netherlands' UB...
Production when garrisoned. Units may cross mountains after the first Shophet is earned, taking 50 HP damage if they end a turn on a mountain. 
gold on coastal tiles, which consumes the unit. Puppets enter resistance when annexed. May purchase in puppeted cities.
on sea tiles worked by the city. +1 
combat strength and is especially strong when defending Carthage's borders and trade routes. 
