As far as Basque is concerned, it is generally understood to be the living descendent of Aquitanian, the pre-Roman language of Aquitania or Vasconia (the area between the Garona/Garonne and the Pyrenees), from which comes the Occitan name Gasconha and its French and English adaptations Gascogne and Gascony. The indigenous language of the Gascons, likely descendants of the ancient Vascones, shows similarities in its phonological evolution to the Spanish dialect of Castile, which through geopolitical chance was imposed on most of the rest of the Iberian peninsula. For /f/ in other Occitan regional varieties, Gascon almost exclusively has /h/: "Lo men hrair que vedó lo son hilh a la frinèsta/hrinesta/hièstra" instead of "Lo meu fraire vegèt son filh a la fenèstra" (My brother saw his son at the window").
There seem to be links between Basque (via Aquitanian) and Iberian, but these could just as well be due to borrowing as to a genetic relationship given the present lack of knowledge of the Iberian language. (Look up "Iberian language" in WIkipedia for a primer.)
There were a couple of other pre-Roman languages spoken in the peninsula: Tartessian and Celtiberian (as far as is known nowadays).
As for the reasons why Latin and then Romance (Vulgar Latin) took root instead of Greek, I think the fact that Latin was THE single official language of the empire probably had a big role to play. Don't forget that Emperor Cicero was from Tarracona (now Tarragona). Several other emperors were from outside Italy as well.