Sarde,
I have a bit more time and energy, to perhaps give a useful response to your questions.
Does that mean all Christians should eventually cut all ties with their family members who are not Christians?
No. Disciples of Yeshua, considering the cost of discipleship, will let the word of God divide them from family members who choose not to become disciples, and will choose to "hate" (as God, not man, defines that word) those family members. They will do this remembering Yeshua's own example, where His mather and brothers, thinking that He was out of His mind, came to "rescue" Him and take Him home. He said that He regarded His disciples as His mother and brothers, meaning that His natural family had to repent of their view of Him, and become His disciples, before He would regard them as family. This they did, and, from the cross, and after the resurrection, He affirmed their family status with Him.
Disciples may also remember Peter's example, where he brought Yeshua home. Yeshua healed Peter's mother, who took care of them. But, we have no further sign that she became a disciple, or received any more salvation.
The point to disciples is that diligent obedience to our Lord's commands gives everyone we know their best chance at salvation from evil in this world, and in the world to come. When God is assured that our natural families pose no threat whatsoever to our love for and obedience to Him, His jealousy is appeased and He will save them so that we may enjoy them further.
Can we not love people who are not Christians?
Yes, we can. We are in fact commanded to love our neighbors, whatever their spiritual position. Remembering, of course, that "this is love, that we keep His commandments." Again, man's definition of love is often God's definition of hate, or worse. The tender mercies of the wicked are cruel, says God. And, it is better to meet a she-bear robbed of her cubs, than a fool in their folly. Men have thought that "the road to Hell is paved with good intentions." The paradox of "hating" someone in order to "love" them properly clearly exists in the spiritual world. That's why He asserts that "wisdom is the principle thing."
I dearly love my family members and even if they believe differently, I don't think they are godless or 'dead in their sins'. I live close to my family members and regularly visit them. Should I move to the other side of the country and say I want nothing to do with them anymore because I am now a follower of Christ and they are not?
No. You should go into your prayer closet, and ask God to speak to you about each and every one that you dearly love. Ask Him how you yourself are doing, and what you can do to further please Him, and keep His commandments. Then ask Him to give more life to those you love, starting with foundational matters such as the will to live, and the choice or love of truth about how to live. And, of course, to have a heart that loves Him, and is drawn to His Son, Yeshua. If He tells you to move, not away from them, but to a place of unity and love with other believers, obey in faith and assurance that such a move will greatly increase hope for those in your family that you love, who may be spiritually troubled. See yourself as going for help, not abandoning those you love.
The church I go to has as one of their goals to have people 'engage in meaningful relationships with unbelievers' in order to try and bring them closer to God.
Satan has the goal of entangling you in meaningful relationships with unbelievers, so that you are crippled in God's service. He also wants you to settle for "going to church," instead of "being the church." In the latter case, you will strengthen the church in its tearing down the gates of Hell. In the former, you will remain separate from the church in some foundational sense, and his (Satan's) for eternity. Don't go to church. Go to your prayer closet. Then the church will come to you.
Jesus says He comes to divide families. Could He be talking about people severing their ties with their families because their family members are expressly anti-Christian? Or could He be talking about the fact that people will go to Heaven with Him and leave their family members behind because they are 'dead in their sins'?
He is very clear that disciples leave their families here on earth, as a part of discipleship. Remember too that "dead in their sins" means dead as God defines the word, mainly separated from Him spiritually. Family members who do not hear God calling them, are not conscious of His presence in their life, are dead in their sins. They can be revived, or quickened spiritually, but it takes prayer, the power of which is increased by numbers praying, by fervency, and by the faithfulness of those praying. That's why "going for help" is the way to see helping those you love who are deaf and blind, dead, spiritually.
I mean, I could actually have a chance at bringing my loved ones closer to God. If I leave them, I can be sure that this will never happen. In fact, if I leave them, they will think that this Christianity is a really awful religion to make people abandon those they love...
This is what Satan would like you to think, what he has worked your whole life, as best he could, to convince you is true. So God tells us not to lean on our own understanding, but to trust Him with our whole heart. Wisdom, Sarde! Go to your prayer closet, and pray for wisdom. "If any one lacks wisdom, let them ask of the Father, and He will give it abundantly, without reproach." Read again those first versus of James.
But, of course, Christianity is an awful religion. Flee from it, into the heart and arms and wisdom of Yeshua. He is alive, and has paid a great price for the love of you, that you and He might have heart to heart talks, and priceless walks. Make His joy complete, by going to Him, not church or Christianity. He takes no pleasure in fools, and hates hypocrisy, and wants you to stay away from places where such gather.
Try it! You'll see. You will, I'm am sure.
Stephen
™ Version 4.2