Hello,

Lets assume, that I have one attacker with only one hit point againts one defender.

An edge battle starts and the defender uses fate card "Heat of the Battle"

Can I use then to prevent this damage (and to prevent killing my character) an event card such as "Our Most Desperate Hour" and put a shield on this unit or Lightsabre deflection and put a damage on the defending unit ?

No and yes. they are two different types of events. one is an action, one is an interrupt.

in the case of our most desperate hour, your action windows are after attacker declares, after defenders are declared, AFTER the edge battle is resolved, and after every strike. so if you wait until after the edge is resolved, that heat of battle cannot be affected by our most desperate hour, and it doesn't interrupt anything. it merely puts a shield on a unit of yours. however, it could have been used in one of the 2 action windows after attackers are declared or after the defenders are declared, but you wouldn't have known about the heat of battle by that point.

light saber deflection, however, IS an interrupt. so it allows you to interrupt damage being done to non-vehicle units. in the same scenario, you wait till the heat of battle is being resolved and then use lightsaber deflection and deflect the damage to your opponents unit instead of yours.