Not only should you be able to get a formal quotation from the hospital but you can shop hospitals as well - if there are others your insurance covers. You may be able to save money at another hospital or talk one down by using the other's quote. They may only be willing to give you the cash price though - make sure you get them to clarify.
If you have insurance you will have to pay insurance, not the hospital. The time frame in which to do so will depend on your insurance, and they will usually work with you to accommodate your needs. Don't be afraid to ask.
Medical billing departments are accustomed to long delays, so they're pretty lax on their terms. They understand that even if they bill you, you're going to submit it to your insurance and have to wait for that processing to get back to them. It's just part of the process.
Good luck and congrats on your soon-coming arrival!