The correct answer for a) is do. I'm having trouble understanding why 'do' should be used to replace 'does'. Shouldn't the correct answer stay the same as it is which is 'does'? Can someone please explain?
Your test is wrong. Gerund clauses acting as subjects always take singular verbs, not plural ones. This is correct as written:
Keeping cool during the hot days does not always require the air conditioner blasting away.