I did a bit more thinking through this issue and I have definitely not answered all these questions.
However, I have written some of my thoughts out.

Should Christians try to push for God’s laws on society?
Actually in some ways, God’s law is already influenced on society. Although, new world views are pushing against this and we are seeing the direct results of that.
God’s laws are part of the founding of North America.
Even our founding fathers, some being Deists, recognized the truth of these laws, because they wanted a moral society. Regardless of whether or not they wanted to be under the authority of the “one true God” or not.
Examples of this are: “Penalties for crimes to be dealt out by the magistrate for murder, rape, adultery and homosexuality, and other crimes.”
People must borrow from the Christian world view, to have an objective moral view. Otherwise everything is subjective.
Can we expect our society to agree with this?
I believe that the answer is yes and no.

Yes, we should try to push God’s sovereignty over every sphere of life. When we look at passages, like Romans 13, we notice that even Government is under the authority of God. The power that the Government does have is given by God.

No, in the sense that this truth will not be completely lived out, until the New Heavens and the New Earth.
The New Heavens and the New Earth however, will be the only real theocracy, made up of the elect from all time.