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[nalak42]

As the others say, you should be fine. You don't need to use the other splats if you're running a game. I would suggest checking talking with the group first just so they know if you're not working with other splats, or that you haven't got/read those books just to give them forewarning or find out if there's a particular interaction they really wanted to get in on. (And as Heavy Arms mentioned to avoid them putting resources into things that won't be useful for your game.)

[nalak42]

Probably what's left of the god's power. Plus the abilities of whatever other gods were necessary to catalyze the exigence as it would be a rare god to have their portfolio completely expunged from reality at the conceptual level and still have enough power to fully make an exigent on the ashes of their former power.