There is zero physical evidence that Carthage was founded in the 9th century, much less enough to pinpoint a specific year. Based on current archaeological data, I am fairly confident that a foundation date around 750 BCE, or slightly earlier, is about right. This places Carthage at the head of the western expansion horizon for the Phoenicians in this period, and is strongly correlated with Phoenician archaeological evidence from other western sites like Pithekoussai, Dona Blanca, and the enclaves in Andalusia like Morro de Mezquitilla.

Your "mysterious lost century" is probably due more to a rift in "western" (read: Greek) and "near eastern" chronologies. Western Phoenician archaeological data has been strongly tied to Greek material with which it has been found (including at Carthage), and Greek material from the Late Geometric down through Late ProtoCorinthian has a very long, complicated establishment of chronological tradition. This tradition, based largely on elaborate pottery typologies, has never matched well with "eastern" material for which there is no Greek ceramic accompaniment. =Neither is right or wrong--they just don't match up quite right, and we don't have enough mixing of western and eastern material in the period 850-700 BCE to "fix" it (each side claiming the other is wrong doesn't help, either). Basically: your 814 BCE in near eastern chronology seems, for the moment, to be equivalent to about 750 BCE in "western" reckoning.

Now, more on what is going on at Carthage. In the late 80s, there was a lot of talk about the apparently large footprint of Carthage even from the earliest date. Some estimates ranged as high as 60 hectares for the settlement by 700, a number which was probably influenced by German excavation site maps. They had done sondages in the late 80s which had revealed archaic material along a ca. 800 m stretch of the Byrsa hill from east to southwest, though rarely did an individual sondage produce enough to classify that area as "residential" or "industrial" or "necropolis" (see Rakob, F. 1987. “Zur Siedlungstopographie des punischen Karthago.” Römische Mitteilungen 94, 333-49). From these findings, the big mainstream writers on Carthage generated maps that showed the city as quite large in the early archaic period (Hence Lancel and Niemeyer, in their maps of “Archaic Carthage,” include the area from the Juno and Dermech hills south to the beginnings of the later Punic ports at Salammbô). All of this to say simply that: by the early 1990s, a lot of material from the early archaic period was coming to light, and it was becoming clear that Carthage was not the same as these small "emporions" of Andalusia. This was a major settlement, from the beginning or at least from very early on, stretching across dozens of hectares and seemingly enjoying some kind of coordinating role in the western med even at the end of the 8th century.

In the 2000s, there was important rescue excavation work undertaken at the Bir Massouda site in the center of modern Carthage by the University of Ghent (et al). You can find these reports in English under Docter et al in the publication Babesch: Bulletin Antieke Beschaving volumes 78 (43-70) and 81 (37-89). They found, briefly, that a lot of the material from the German sondages was indeed early archaic, but was in fact extra muros (outside the ancient walls) and not a part of the settlement itself. Much of the material they found at Bir Massouda was related to industry and forging, which would have always been at the edges of a Phoenician settlement in this period. They also confirmed the one and only 8th century necropolis during this excavation yet discovered, another feature which was without exception at the fringes of a settlement. The necropolis featured pozzi style graves (pits) cut into bedrock, which would have housed cremation urns. We find nearly identical pozzi graves in the same period in other Phoenician enclaves, including at Tyre back in Phoenicia.

The material immediately above (and therefore more recent than) the pozzi graves contained the typical rainbow of western Phoenician settlement material: imported transport amphorae from Andalusia, Levantine Phoenician red slip with good typological comparanda, Corinthian fine wares, and a lone Athenian "SOS" type transport amphora. In a similar, but not directly connected trench, was found a Euboean Late Geometric skyphos. All of this amounts to an avalanche of evidence for a date ca. 735-700 BCE.

The take-away is this: no, the site wasn't quite as large as we had thought in the late 80s. Nevertheless, an estimate of 15-25 hectares is feasible, and that is still a gigantic footprint for a settlement 750-700 BCE. We see evidence of rapid expansion as our lone 8th century necropolis is "pushed" beyond the sight of our excavation, probably to make way for the expanding city's residential and industrial needs.

Bibliography for this topic is piecemeal and pretty complicated, unfortunately. The German excavation reports are all in German, as is their wont, but Docter's summation reports for Ghent's rescue dig mentioned above are English and relatively accessible. When dealing with early Archaic Carthage, it is dangerous to take any source before Docter's publications at face value. It was (and is) a fast-changing area of understanding.