Mat 5:43 ff implies that God loves everyone. We are told to love our enemies, because we're doing what God does. Admittedly it doesn't actually say that God loves his enemies. It says that he does good to them, but not actually that he loves them. But the passage makes no sense unless he does good to them because he loves them.
This needn't mean that he loves them in the same way, of course.
It seems to me that we have passages that say God loves everyone. (No, I don't think "all kinds of" can be stuck in front of most uses of "all" in the NT.) But we also have passages talking about him loving his own. The simplest way to deal with this is to say he loves them differently.
If this quote is right
Calvin and Calvinism » Blog Archive » John Calvinâs Doctrine of the Grace of God: General and Special, then Calvin took a similar approach to 1 Tim 4:10. He understood it as actually meaning everyone. God is in some sense a savior to all, but not in the same sense is those who believe.
What Calvin does in John 3:16 is fairly subtle. He does not try to understand "God loves the world" as "God loves part of the world." He says "the Heavenly Father loves the human race, and wishes that they should not perish." It's an interesting approach, because saving the race doesn't require saving all members of it.
On the other hand, Calvin does sometimes use "all means all kinds," e.g. in 1 Tim 2:4. He evades the issue in 1 Tim 2:6, but implies that he thinks all means Jews and Gentiles.
Calvin doesn't deal with this in Mat 5:43. He says we should imitate God's kindness, but never explicitly says that God loves his enemies.
But whatever Calvin says, it seems clear that Jesus did have that in mind. I would use Calvin's approach for 1 Tim 4:10, and say that God loves everyone, but especially those who believe.