I know I am not Steve, but I can say that you may get different answers on these questions by different Baptists.

"1. Since you propose that only professing believers are in the New Covenant, how does that relate to visible church membership in your church?"

As you probably are aware Baptists who hold to the LBCF, believe that we can not fully know who is saved and who isn't. We baptize not because we know the person's heart, but because they profess to be believers and some fruit is showing.
I particularly believe (not all Baptists agree) that membership to a Baptist Church should not be on whether or not someone has been baptized as an adult professor. But on the basis of whether or not after the elders are relatively certain that they are Christians and are willing to commit themselves to the well being of the rest of the Church body.
This would include Christians of Paedo-Baptist persuasion. However, since they disagree with Credo-Baptism, they would not be able to serve in the offices of elders or deacons.

"2. Do you view your children as children of the church or are they children of the world?"

Before a child has made a profession of faith, they should not be presumed to be Christians. But that does not mean there are not advantages to a child with Christian parents in the Church.

4. What advantage does a child have growing up in a Christian household?

Besides the obvious benefits of the parents and the Church doing everything in their power to make sure that child knows the ways of the Lord.
In 1Cor. 7:14 we read " 14For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy."

This of course does not mean that children are holy in themselves. It means that they are set apart, because of the believing parent. They are privilaged because at least one parent is a Christian.

"5. How do you view infant dedications?"

I am not sure there is an official stand on infant dedications in the Baptist Church (I am going to look for one).
But my understanding is that when a baby is dedicated, it has more to do with the parent/s and the rest of the body dedicating themselves to bringing that child up in the ways of the Lord.

6. During a typical infant dedication do you pray for God's blessings on the child?

It has been a while since I have seen a baby dedication, but from memory the praying had more to do with asking God to help everyone to bring the child up in the ways of God.

7. Does the church assume some responsibility for this child?

Hopefully I have adequately answered that already, the answer is yes.

I will add one question.
8. Are baby dedications mandated in Scripture?

Not sure, but I think it is safe to say that it takes a whole Church body to bring up a child in the ways of the Lord.