1° You ask your armature to play arm action when collision … but your Armature physics is set to “No collision” . So its normal it doesn’t work
2° It’s better to keep the Armature on “No collision” and keep smithgameenginebox with Rigid Body … so it’s smithgameenginebox that will send the pulse to the Armature. As you can see, you forgot to put Arms action Actuator to Level 1
Here, i corrected for you
3° As i said , you don’t want to have your legs animated in your Arms animation (unless you want replace the leg animation)
Reminder : Action actuator with higher level is played on top of others.
Demo :
https://imgur.com/a/HHPyvlA
(i reanimated arms action to make it more obvious)
btw, Collision should not be the thing that kills you … look , you are immortal if you don’t move 
https://imgur.com/a/JSENiQd
What you want is to lose health each time the monster hits you … Here’s how to do it
In the video I demonstrated layer 1, but nothing happened.
As for the other changes, I don’t know.
The problem is, the AI doesn’t get removed or a new scene loads, as for the AI to the player that works. I can’t quite get it to click yes for it to restart. Needs some sort of code for the cursor.
Hmm, if the FP collides with the AI, the health goes down, that shouldn’t happen that way, only with the bullet.
It isn’t that great, because the bullet bounces off the AI. So it flys up.
I did check invert on the actuator brick, I didn’t do that in the previous clip.
And the armature was on no collision. Only the AI cube has collision. I wouldn’t of gone wrong there with this. 
You forgot this link and you have to put the right mode
https://imgur.com/a/epMctmU
made with BGE 2.79
lischaracterSep10.blend (2.3 MB)
Neither that file or mine actually cause damage with the bullet. Only the empty colliding with the AI causes the damage.
So this type of program setup isn’t good.
The example by vill89 was a python code one, which was a better idea, it would seem. I am not sure now. He may know best on this.
The animation in you’re copy is too rapid. Eitherway, the program is the problem.
Thanks
