1. Well, okay. Before the unification of Dorne with the rest of the Seven Kingdoms, what did they do for a High Septon? Was there an Antipope-type of rival supported by the Martells? An Archsepton of Sunspear? Or did Dorne accept the High Septon, despite being based in King's Landing and beholden to the Targaryens?
The Faith is not limited to the Seven Kingdoms proper, any more than the medieval Church was limited to Italy. So no, no Dornish antipopes. And the High Septon was not especially "beholden" to the Targaryens. In fact, he was a dangerous rival... well, until the reign of Maegor the Cruel, at least.
I suppose this leads into the larger question of who controlled the High Septon prior to the Targaryen Conquest.
See preceding answer.
2. You've mentioned that Dawn has an illustrious history -- is there a ballpark figure for how long the Daynes and/or Starfall/Dawn have existed?
Oh, I'd say Dawn goes back a couple thousand years... and before that, things get a little fuzzy anyway.
3. You've also mentioned that properly, saying just "Martell" isn't accurate, and pointed to Oberyn and Doran beying listed as "Nymeros Martell". Are the women of the line also "Nymeros Martell", or does the Nymeria-reference take a different ending to indicate gender?
No. Nymeros indicates "of the line of Nymeria."
4. Do all Houses of Dorne follow the first born heir law?
The vast majority, yes. May be a few stony Dornishmen in the mountains who go their own way, those least touched by the Rhoynar.
5. Is the spear the chief weapon of choice in Dorne? And what type of armor do they wear?
Lighter armor than elsewhere in the realm, due to the heat. And yes, they use several forms of spear (not just the long spear that the Red Viper employed against Gregor), but also swords, lance, axe, and the rest of the usual arsenal.