S. -
Every hospital/doctor is different, but I can answer you based on my experiences (I have three kids).
Do they take the baby away from you after he's born? They take the baby away to put them under a lamp, clean them up, weigh them, do the APGAR score, etc. They usually will let you hold the baby either right away if all is well or after they are done cleaning them up. Then depending on how the baby's doing, they'll take the baby for a short period of time to run some preliminary checks to ensure all is okay. The time varies though - with my first, she was away for 3 hours as her temperature was very low - they put her under heat lamps and it took a while for her to warm up before I was able to get her.
Can the father stay with the baby during checks? Yes, absolutely. Every hospital has different rules on the amount of people allowed in the delivery room, but I've never heard of them not allowing anyone in the room, especially the father. If you get an epidural, they may ask the father to step out while it's inserted though.
When can you go home after a normal delivery? For a normal delivery, they usually release you after 2 days (assuming you are recovering okay and it was a vaginal delivery).
If you have to stay, does the baby stay with you in your room? This depends on the hospital. There are some hospitals that allow you to have the baby in a bassinet in your room. In that case, it's up to you whether you want the baby with you or in the nursery. With my third, I chose (at night for the first night) to have the baby in the nursery for the time in between feedings (I was nursing), so they would bring the baby in when he was hungry and then I could rest for an hour or two before the next feeding.
How does it work with painmedication? I'ts your choice whether or not you want them.
Do they wash the baby after birth? Yes, absolutely.
Can the father cut the umbillical cord? Again, depends on the doctor/hospital, as well as the state of the baby after the baby is delivered. IF there are issues with the baby, then there likely won't be time. IF all goes well, the doctor should ask you. My husband cut the cord on the first two of our children. HOwever, with our third, the doctor didn't even ask and just cut it. This is something that you can and should discuss with your doctor before delivery.
I also read all kinds of things about a birth plan, what is that and what do I do with it? I think it's just so that you're prepared and know what to expect from the doctor and the doctor from you. I believe you'd include in it whether or not you want an epidural (medicine), who's in the room with you, if you want the father to cut the cord, etc.
I hope this information helps you out. Best of luck getting situated in the states and with the rest of your pregnancy!!
- K. :)