We know very little, but I'd like to congratulate you on the quality of the web pages you've managed to dig up: the piece by Debiasi refers to a scholarly piece (the Cambridge Companion to the Epic Cycle) that isn't in print yet.

Another of your links goes to this site, which contains a complete online copy of a recent (2003) edition of the surviving fragments of the Theban epics, along with an English translation: which is very cool, by the way, I didn't know the Loeb Classical Library were doing that! Anyway, if you look at pages 38-63 of that edition you'll see virtually all of the ancient testimony about these poems that survives. It's not quite perfect, as that edition is somewhat condensed, and not a critical edition, but it's good enough that it makes little difference. If you read that, you'll know about as much as anyone.

We know enough to know that they weren't actually a "cycle". The only evidence of them being grouped together is one of the so-called "Iliac tablets", the Borgia plague, from the 1st century CE, which lists the Oidipodeia, Thebaid, and Epigonoi as a group. Another source that draws on material going back to the 4th century BCE, the Contest of Homer and Hesiod, refers to the Thebaid and Epigonoi as a pair. They weren't a cycle and their grouping together is largely a modern convention based on their linked setting at Thebes during the time of Oedipus, his children, and grandchildren.

Based on what we know about the myths, and the hints contained in the surviving fragments, we can at least infer what each epic was about. The Oidipodeia covered the story of Oedipus; the Thebaid covered the war between his sons Polyneikes and Eteokles, and the "seven against Thebes"; the Epigonoi covered their children's more successful campaign against Thebes, led by Alkmaion; and the Alkmaionis covered Eriphyle's murder of Alkmaion. These stories are given in greater detail by Hellenistic mythographical sources like the Library of pseudo-Apollodoros, but you should also be able to find them in any halfway decent encyclopaedia of myth.

We also have testimony on the length of the first three epics: according to the Borgia plaque the Oidipodeia was 6600 lines long, and according to the Contest of Homer and Hesiod, the Thebaid and Epigonoi were both ca. 7000 lines long. For reference, that's a bit over half the length of the Odyssey (12,110 lines).

They weren't terribly influential poems. We know of dozens of heroic epic poems dating to the Archaic period, but the two main Homeric epics are the only ones that enjoyed great prestige and survived to the present. They died out for the same reason that the Trojan cycle did (which was a cycle, published as a group in antiquity): they were antiquarian curiosities, not literature that continued to have an impact after the time in which they were composed.

It's difficult to gauge how much popularity they had in their own time, but the Thebaid at least was popular in the 6th century BCE: there's one Attic vase that survives showing a rhapsode performing a line from it at the Great Panathenaia, indicating that it was regarded as one of the canonical Homeric epics; and Kleisthenes, the early-6th-century tyrant of Sikyon, banned the performance of the Homeric Thebaid because it celebrated his rival Argos (Herodotos 5.67; Herodotos doesn't specify the Thebaid, but no other known poem fits the context).

The 300-page Greek document that Wikipedia cites is a critical edition of epic fragments from 1877. It's there because Wikipedia strongly prefers obsolete sources that are online to up-to-date print sources. Still, as out-of-date editions go it's not half bad, and if you want the surviving fragments of the Theban epics they're on pages 8-14 (Oidipodeia, Thebaid, Epigonoi) and 76-7 (Alkmaionis). But the 2003 Loeb edition is likely to be more up your alley.