What's the point of supplication prayers when the living believers are asking the other living ones to pray for them? I mean apostle Paul asked other believers to pray for him. Why did he do that while he himself could pray for himself?
I think prayer (especially supplication prayer) is one of those things that can't be removed from the context of the Gospel, and then properly understood.
First, God desires to have a relationship with us. This is the whole reason he sent Jesus. He loves us enough, even though we are born with our fist to the sky against him, to send Jesus to die. It's against this context that we look at prayer. If he loves us this much, doesn't it stand to follow that he wants to have a relationship with us? (The answer is yes.) If this is the case, then praying for another person isn't a burden, it's a privilege and a joy provided only to those who can approach the throne of grace with confidence. (Hebrews 4:16)
Next, in this relationship, he wants us to acknowledge his complete and total provision for us. He, being the creator of everything, has, by definition, given us absolutely every good thing we have (James 1:17: Every good and perfect gift is from above, coming down from the Father of heavenly lights, who does not change like shifting shadows.) It's in this truth that he desires to be glorified, as a God who provides all good things.
Prayer must be seen as more than simply "what we can get by asking", because God gives us several things that we never ask for, such as breath when we're young, friends, sunshine, etc. So if that's the case, it leads me to a couple of conclusions: First, God desires to bless us (Matt 7:7) (I walk a thin line here... I'm not talking about "prosperity Gospel" here. He gives us what he knows is best for us, not what we think is best for us.) And second, God asks us to pray in supplication, because he wants to bless us, and he wants to be seen, rightly, as the one who blesses. It's a way he can build our dependency on him.
Ask any dad. They know what their children want before the ask, but they want their kids to ask, so that when the provide their children with what they want, their children are thankful for the provision. If the provision is given without the asking, the gratitude is far less likely to happen, because we're inherently selfish people. Christianity is like growing up in reverse. Instead of us learning to be more independent, we learn to be more dependent upon God. The strongest Christian is the one who is the most dependent on God, and least dependent on self.
Now for the meat of the question: Why should other people pray for other people, when we can pray for ourselves. I think this again comes down to relationship. Praying for someone else means you set aside your own selfish thoughts for a while, and think about somebody else. Praying for somebody else means you come alongside that person and acknowledge that you're "on their side", and you too would like to see God's provision for that person's life, and you're likely to rejoice with that person when God does provide. It's a way to communally love one another and share in someone else's sufferings.
So prayer is more a tool that changes the one praying than it does the circumstances around the prayer. It leads us into dependence on God, and into community with one another. It binds the church to God, and binds the church members to one another, encouraging unity within the church. Paul's mission, and ours as well, should therefore do more than simply spread the Gospel to the unbelieving, unknowing world, it should also help to grow the Gospel within those who believe. Supplication for others helps that.