OP, you ask a really valid question I think most believers and unbelievers ask.
We pray because the word of God (bible) tells us over and over that we should. Anything that God bothers to say once should be taken note of, but anything that God says repeatedly more so. He obviously really wants to get the message of communicating with him across, loud and clear! In the same way that we communicate with our partners, we should communicate with God to deepen the relationship. We don't expect a great relationship if one person refuses to talk to his partner, but we expect to feel closeness and a sense of oneness if we communicate really well and really often - and so it is with God. Any parent with children living away know the joy of picking up the phone and hearing from your dearest. God is just the same.
Jesus himself prayed a lot and is an example of reasons to pray. God loves to hear from us! Remember he prayed that God would pass the cup of suffering from his lips, and yet he still suffered. Sometimes we pray that God will stop our suffering and we find things improve, but other times things don't. We know that Jesus had to suffer to go to the cross, and we know that God's plan was to offer that sacrifice for all sins so we don't have to pay our own sin-debt, but we don't always know what God's plan is in this life and so sometimes it doesn't make sense to us.
With regard to refuges, the tragedy of these people suffering are due to the wickedness of their fellow-man. God has a plan for dealing with wickedness. None of us are perfect, we all have our faults, but we often see it as faults to differing degrees. God says that all faults result in the consequence of punishment, and so it stands to reason that, if He forced people to be perfect then He would have made robots, but obviously chose not to.
If God killed all ISIS dead tonight because of their wickedness, then to be fair he'd have to wipe out most of the rest of us, because we've all hurt someone at some time in our lives. Sometimes badly. Sometimes more to ourselves than anyone else, but nevertheless, it's still human badness. But God has more mercy than that, and offers people a lifetime of choice until it is then too late. The door is shut because the person has died either believing in Him, or not. Their choice. remember God hasn't made robots.
But that doesn't stop God nudging people to make responses that help. They choose to respond to that nudge, if that makes sense. They could choose not to, in which case God could just as easily nudge someone else to do His work.
Prayer does change things. It changes our heart response, it makes us closer to God. It gives us hope because we are reminded as we pray "your will be done" that there is a bigger picture and a plan that we might not always understand at the time, but looking back it all makes sense.
But in the same way that if you refuse to communicate with your partner, you can't expect a decent relationship to flourish, being distant from God is like having a moat around the castle. In order to drop the draw-bridge and gain access, we need to be in a right relationship with God, which means to the unbeliever to trust in and rely upon God (rather than themselves or other gods of the age, like money, power, status), and to the believer involves confessing sins and staying pure so that God responds to your prayers.
Remember he's not a 3rd class genie though. It's a bit like your 3 year old asking for sweeties and sometimes you give it, but other times you say no because you know a diet of sweets alone won't help them grow well. Not all 3 year olds will understand that plan until they're older.