What is Faction RTC? If I use it to buy items in-game, will those purchases count towards the $20 US required to unlock the premium game?

Faction RTC is rewarded upon completion of a campaign. Faction RTC is only rewarded once per campaign per faction, therefore running the same campaign with multiple characters will result in only the Requisition points rewards on subsequent characters.

Faction RTC can only be spent by characters of that faction. Faction RTC is spent first before regular RTC. Since items purchased with RTC are account bound this means certain cosmetic items purchased with faction RTC will be available to multiple characters if they are of the same faction.

What this means is that, indeed, Faction RTC rewards do not count towards the "pay $20 to become premium".