I keep seeing those round compass-like red things drop during battle. Aren't those supposed to be star coins dropping from the enemies? Yet my star coin count remains at 0. Am I wrong about what those are?
Those are actually clocks, and they reduce the Cool Down period of your main Eidolon.
Star coins are purchased with Nutaku Gold, which in turn is purchased with real money. So unless you want to spend some real money on the game you won't get any star coins. Just have to patiently wait for Magic Jewels.