Correct :
1) Did you see my dog? He's run away .
2) Mandy will buy a car as soon as she will pass her driving test .

Passes*

1 Have you seen my dog? He is run away.

2 Mandy will buy a car as soon as she passes her driving test.

1) Have you see my dog? He has ran away. 2) Mandy will buy a car as soon as she pass her driving test.