"heat of the moment" is an expression related to strong emotions like passion or anger. In the heat of the moment, he kissed her. In the heat of the moment, she slapped him. "on the spur of the moment" is an expression related to impulse or a decision made at that moment. She decided to go to Spain on the spur of the moment. He decided to buy the model plane on the spur of the moment.

In the heat of the moment is more in an argument and on the spur of the moment is more a quick decision to do something. Like: In the heat of the moment she decided to throw away the note from the friend with whom she just had an argument with. But: On the spur of the moment she decided to drive to the beach as this was her favourite place to chill out.

Yes they mean the same thing.

Exactly the same! It's up to you to decide which one is the best. :)

Yes, both mean the same: while tomporarliy angry, excited or engrossed, and without stoping for thought.