Lastly, I assume that Christians respect the laws of logic and therefore their justifications must also follow these laws.
Then you're not really going to get very far. Yes, Christians respect the laws of logic, but we do not respect only the laws of logic. The basis of Christian belief isn't grounded in logic, but in testimony, faith and experience. We know that God is there because he makes his presence known through the witness of the Holy Ghost, and we know that he is good and worthy of obedience and worship because of the history of our relationship with him.
As we follow God's commandments faithfully, we can see the results in our lives. We see that living the principles of the Gospel works, that it produces positive results, helping us to resolve or avoid problems in our lives. That increases our faith, making us better-able to trust in God, so that when we need to do something truly difficult, even if it seems to contradict what we think is logical, we can say, "God has never led me wrong before, so I can trust in him now." Those who don't understand this principle say that Christians live by "blind faith," but that is far from the truth.
As for how Christians convince others of their belief, we don't. While it's possible to convince another person that our viewpoint makes sense, actual faith, the kind I'm talking about in the previous paragraphs, comes from God, not from another person. What proselyting Christians attempt to do is convince others to open their hearts, to be willing to accept that the Gospel might be true, and to be willing to pray to God to seek their own testimony and establish their own relationship with him. Conversion is a highly personal matter between oneself and the Lord; anyone else's involvement is secondary.