1
My take is that there is no way one can prove the existence of a Judeo-Christian God by logic or science.
In fact, I am not interested in this approach. The writers of the Bible cannot have any idea of modern logic or science, and this approach seems to be shoehorning clear-cut ideas into a complex term with much historical baggage.
Instead, the most we can do is to show that the concept of a Judeo-Christian God is intelligible and not self-contradictory.
My favourite argument on this is from Herbert McCabe: at its historical core, God is understood as the reason behind everything. Thus, the concept of “God” is as good or bad as the concept of “reason behind everything”: something not obviously vacuous or self-contradictory.
2
Even though Christianity developed in an intellectual atmosphere that is alien to modern science, there is one core claim that has an empirical, falsifiable favour: that Jesus, a historical human being, died and came back to life.
C. S. Lewis has argued that this story is very odd in the context of the Jewish tradition around the time of the historical Jesus. I do not know enough to form a judgment: but I do find it striking that none of the Chinese belief systems I know is dependent on a factual, falsifiable claim.