Yeah, the husband can stay if it's a private room. However, they don't usually have the private room available. Most of my friends could only get the private room on the last day or so, and so did I.
However, if you could complete the labour and delivery as early as possible during the day, you have a higher chance to get the private room. If you complete the labour and delivery at nite, then the chance is low.
You can get the 乙肝针 with your paediatrician afterward, or you can tell the hospital to request for it. If you were born in China and the child has a chance to visit China, they will give the 乙肝针.