S.W.
I delivered both my babies at Baylor Grapevine, and I had great experiences both times. I delivered naturally with both, and I never received pressure from the doctors or nurses to go for the drugs. I had fairly easy labors and no complications so I don't know if that would change things. They did put the IV in my hand, but it wasn't hooked up to anything. It was more of a precaution in case things went south and I needed to be rushed into surgery. That didn't happen so they pulled the needle out right after birth. You can ask, but that might be hospital policy due to liability in emergency situations. I dont know what EFM is so I can't answer on that.
As far as birthing positions, I think they'll listen to requests on that too. They were going to get me a birthing ball or let me sit in the tub, but my labor ended up going too quick for that. With my first, they left me alone for a while and only came back to check on me every so often so I would think you can get into whatever position you want. If you're not hooked up to an epidural, you can move around as much as you want.
They handed me my babies right away. I got to hold them while they finished stitching me up etc, but then they took the baby to clean him up...I have a friend who delivered there more recently, and she said that they now clean the babies up in the birthing room so dad ( and whoever else may be in the room) is with the baby the whole time.
The nurses there are great, and very accommodating. They do whatever you request ( as long as it is within reason and not of danger to you or the baby). I chose to have the nurse take the baby at night to the nursery so I could try to sleep, and the nurse would bring him back in every few hours to feed him. They never tried to push formula or a pacifier.
I can't say enough good things about Baylor Grapevine. If I have another baby, I will go back in a heartbeat.
I hope this helps. Good luck!