[assembly] How do AX, AH, AL map onto EAX?

no your ans is Wrong

Selection of Al and Ah is from AX not from EAX

e.g

EAX=0000 0000 0000 0000 0000 0000 0000 0111

So if we call AX it should return

0000 0000 0000 0111

if we call AH it should return

0000 0000

and when we call AL it should return

0000 0111

Example number 2

EAX: 22 33 55 77
AX: 55 77
AH: 55    
AL: 77

example 3

EAX: 1111 0000 0000 0000 0000 0000 0000 0111    
AX= 0000 0000 0000 0111
AH= 0000 0000
AL= 0000 0111