1. What happens if you attack an enemy and you do not kill it in the attack phase. Do you end your turn and continue the fight during your next turn?
It depends on the situation. In a city, keep or mage tower you retreat to the hex from which you entered; following the rules for forced withdrawal if you managed to enter from a unpassable hex. Elsewhere you remain where you are. What you do on your next turn is up to you.
2. If you gain a crystal token in your inventory can you do the following -
a. Use a crystal from the inventory AS WELL AS a mana die from the source during the same turn?
b. If you do not use a crystal during your turn, do you discard it? Do you discard it after the end of the round?
A crystal to your inventory you keep. A mana token is lost if you don't use it on that turn.
3. If you end your turn on a village, at the start of the turn can you still buy units? Or only when you land on a village?
You may use influence to recruit units on the following turn but you wouldn't then be able to move away from the village that same turn as you would've given up your movement phase and jumped directly to perform an action.