God always answers prayers, it's just that sometimes he answers, 'No.'. We aren't insignificant to God. Maybe our time on Earth is insignificant in the whole scheme of things and that's why it's easy for God to say, 'No.' in certain cases.
You are right that we should live the life we have. We should also treasure every moment with those we love because they might be gone tomorrow.
I think God can, and probably does, answer prayers for strength or courage in the affirmitive. Things that make you a better person. That would serve a purpose important to Him.
But the conditions of this sinful world are such that bad things happen. They happen to everyone and they always will unless and until the prophecies of Revelation are fulfilled and there is a final judgment.
But I'd never see a doctor if I thought the power of prayer was sufficient and effective. None of us would.