1. "Have you ever killed a cockroach?" We use the "Have you ever...?" pattern to ask about a person's experience up to now.
Cockroaches in Australia are massive. We don't so much kill them as face them in mortal combat.
2. "Did you ever killed a cockroach?" This is incorrect. You only need one past tense per statement. "Did you ever kill a cockroach... ?" is the correct tense, but you need a period of time in the past. "Did you ever kill a cockroach while you were in Australia?"
Not so much kill as face in mortal combat.