If three different unit types attack one defending Crystalline, does the defender get a chance to roll the dice each time it is attacked?
No. Each unit in a battle space fires once in the battle.
Does the defender get to choose any opposing unit if successful?
The player who loses a unit in combat gets to decide which unit it is with the single exception of the case where a lava leaper rolls 5 or higher.
If both players are successful, do they take each other out, or do they each get a choice?
Each gets to choose which unit is killed (per above question). The active player must decide first.