Important questions. I have a few thoughts on the subject:
1) Jesus spent a great deal of time in prayer with the Father, therefore His example would indicate that prayer is valuable and even essential for the Christian, regardless of what the outcome (from our perspective) of that prayer might be.
2) I have often struggled with very specific prayers, especially prayers for healing, because we don't tend to see a lot of overt, supernatural miracles as were common in Jesus' time and with His disciples. Perhaps this is a result of our lack of faith, but I tend to believe that God is simply working in different ways now than He was when Jesus walked the earth. God's miracles and supernatural power were needed at that time to prove to humans that Jesus was the Messiah. Now we have scripture and the stories of Christ's life to base our faith on. Jesus even said in the parable in Luke 16:31, "If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead." So it stands to reason that physical healing may not always be God's priority. He's more concerned with our spiritual condition. Doesn't mean I wouldn't pray wholeheartedly if someone I loved got sick...I just think I wouldn't expect or require healing as a proof of His love for us.
3) The reality is that there are many unseen spiritual battles and factors going on that we don't see and may never understand, but God does tell us to pray for one another. "Therefore confess your sins to each other and pray for each other so that you may be healed. The prayer of a righteous person is powerful and effective."
James 5:15-17. So even though we don't understand it all, I say let's just keep praying anyway and let God make the hard decisions of how to answer!