The correct translations are, respectively:

Il aurait pu beaucoup mieux faire à l’examen.

It’s correct to put “faire” before “beaucoup mieux” but better style to put it after.

Je pense que Pierre pourrait avoir tué le chat.

“peut” is correct but more unusual. It implies more certainty. “pourra” would imply you’re talking about a situation where the cat is not dead yet, but is predicted to die.