Won't be able to answer your question with the amount of space the tagboard offers. So let me try answering your questions with this post. Firstly, thank you for asking, please ask more!
But I have to add that I cannot claim to be able to answer everything. And so, all the more I need to read the Bible and try to understand what it teaches.
You asked, "If the Gospel are for preaching to non-Christians, then why do Christian bring Bibles for?
Let me try answer in a rather abstract manner. Suppose you are a Muslim, would you want to study or read the Koran to learn more about what you believe in? Or put it even more abstractly, look at researchers, like math professors, do they not consult textbooks all the time? Or if you are a teacher, do you not read the textbook or at least the syllabus before you teach?
To put it very simply, Christians carry Bibles so that they can learn from it. More than that, so that we can study it and apply it in our lives. But I cannot put it better than what is said in the Bible:
"All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness, so that the man of God may be thoroughly equipped for every good work." - 2 Timothy 3:16-17
The Gospel is not just for non-Christians. It is for Christians as well. It is not just a story, it serves to remind us of the sins we have committed and the need to pray for forgiveness everyday. That was basically the subject of the previous post.
Your second question, "What is it in Christians that non-Christians don't have?? Is it faith??"
Well, physically we are exactly the same. We have a heart, a brain, internal organs etc, we are born and we all die. Emotionally we go through the same emotions, we cry, we laugh, we go through heartaches, we get depressed just like everyone else.
If it is not a physical or emotional difference difference, then what kind of difference is it? You brought up faith. And since you mentioned faith, I take it that you believe that we human beings are capable of expressing faith in some manner, just like we express something like love.
What is faith? Is it emotion? When we say we have faith in our friends, how do we express it? By actions? Yes. By our emotions? Definitely. But then, this is a very tangible aspect of faith. What happens when we consider something we cannot see, like a deity or some "higher power". Then faith cannot just be expressed through emotion or physical works. It's beyond that, and if you believe in a god, then I believe that this faith in a god is spiritual.
But does faith per se distinguish Christians for non-Christians? I don't think so. Muslims place their faith in Allah, Hindus place their faith in their multitude of gods, some Chinese place their faith in their ancestors, animists place their faith in animals and objects like trees. Christians place their faith in Jesus Christ.
What distinguishes us is the comfort and knowledge that we are saved and are assured of life after death. If you studied your moral education, you would know that Christians believe in life after death. (At least this was in my education when I was in secondary school). Who can attain this "eternal life" then? Well, anyone!
Would like to refer you to a passage in the Bible, Hebrew 11. It's too long to cut and paste here, so please click on the link below to Bible Gateway and type "Hebrews 11" into the search field.