Clerics cannot fall though.
If a Cleric works against the will of their deity, yes they lose their faith, but they can simply pick a different faith that supports their actions.
Yes they can.
Clerics get their abilities directly from their deity.
If they go against the will of their deity, they lose their powers full stop. Since their powers are a gift actively granted by the deity.
They can attempt to pray to a new deity, who *MAY* grant them powers, but it's not guaranteed. Nor will it necessarily be the same powers they receive (As each Deity will only provide specific domains to their followers - I.e. Shar will not provide Light or Life domains just as Selune won't provide Death or Trickery domains)