This is the one point in your post that I disagree with. I've seen many non-Christians delivered. In fact the Gadarene demonaic was not a Christian yet Christ delivered him. For that matter, no one who had devils cast out of them in the NT was a Christian. They got saved AFTER the devils were gone. (and as seen before we will do greater works that Jesus did)
The trick is not in delivering the lost but in keeping them delivered (as Luke 11:24-26 applies)
I've heard of that too, and I do need to clarify: If the person is in their right mind and capable of making a decision of coming to Christ, they need to do that first. Why? Because if you deliver them and they still refuse to come to Him (like the witch in my earlier post), they end off much worse than if you had left them alone until they were ready to accept the Lord.
In some cases, the person is mostly in their right mind, but the demons are blinding them to the Gospel so that they can't understand it or else preventing them from actually saying a prayer of repentence. In the former case, I would bind the spirits against interfering, but hold off on the deliverance until the person understands and is ready to come to Christ (if they ever are). In the latter case, you can go ahead and deliver them if the Spirit prompts you and you can see that they clearly want to make that decision but are being held back.
In the case of people that have been driven completely insane by the demons (which I have to stress is very rare), you have to let the Spirit lead. In such cases, I would still prefer to bind the demons so that I could present the Gospel to the person while in their right mind, lead them to Christ, and then cast the demons out.
Cases of Multiple Personality Disorder (which is probably what the victim of Legion was suffering from) are more complex, since the person is in need of trained counseling and inner healing in addition to deliverance, but that's a whole 'nuther subject that would take us off track in this discussion.