I think the relationship you build/maintain while praying should be the main reason for prayer. It's really that simple.
If we are Christian, believing in God, then we should get to know Him. To do this, we can read the Bible and pray. These are our most direct links to God.
It seems to me the first few points you have are kind of indirect... I think the main thing is that to talk with God we pray. We should do this constantly (as written in the Bible) so that we have a good relationship with God. A good relationship with God generally means that we will sin less; God will be able to help you much more in all areas of your life.
As for prayers (specifically requests in prayers) going unanswered, we must remember:
-Prayers are always answered. However, "No" is also an answer.
-Some prayers are answered "yes" - but in God's own time.
-Is the request something that will fit in God's will? Is it made from a Christian heart or a greedy mind?
"- what are things you pray about most often with the most resolve in all seriousness without any breach of conscience: those are probably the most important things in life."
I wish to disagree with this. Even if you meant "most important things to you in your life", this probably isn't a good measure. I might be misunderstanding what you're trying to say. One might be serious about X and almost never pray about it because there's little or nothing to say. Or, the opposite, one might pray for an enemy out of love - not because that enemy is an important aspect of that person's life, but because that person feels compassion for his/her enemy.
I like the message from this video (on the subject of praying):
If you think of praying as sitting down with Jesus and talking - as friends just as much as servant to Master, I think it adds meaning to prayer. You aren't just sending out random thoughts into space, you're actually talking with God on a personal level.
Hope this helps.