If Prophetic Flamespeaker deals combat damage to a player multiple times in a turn (if it wasn't blocked, for example), its triggered ability will trigger each time.

The card is exiled face up. Playing it follows the normal rules for playing that card. You must pay its costs, and you must follow all applicable timing rules. For example, if it's a creature card, you can cast it only during your main phase while the stack is empty.

If you exile a land card, you may play that land only if you have any available land plays. Normally, this means you can play the land only if you haven't played a land yet that turn.