Does the Army pay for your ticket back home once your first child is born (when you are on deployment)?

Depending on the length of the deployment you will get "mid-tour" leave. The cost for travel on mid-tour is on the military. However, you are not authorized leave just because you're having a child. If your mid-tour falls during that period then great, congrats. If not, sorry, the military doesn't have to work around your schedule. I would talk to your Chain of Command and see if you can get some kind of preference for mid-tour scheduling. When I was a training NCO, it was my office's SOP when scheduling mid-tour leaves to have a list of when people planned to have children born or special occasions like anniversaries and such that they wanted to be home for and we worked to accommodate as much as we could. It didn't always work but we tried.

I was in the Marines. But I was on a float with the 24th MEU when my twins where born and they got me to Norfolk, after that I was on my own to get to Oregon.

No, the Army will not pay for the ticket home, and barring complications, might not grant leave either.
Having said that, when my Soldiers had this situation arise, the NCO fund came through with a nice contribution.
But officially, no.