People save up for having a kid the way they save up for anything else expensive. Also most hospitals are 'kind' enough to allow you to set up a payment plan so you can pay it off over a few years.
No, normal case is your or your husband's insurance pays for it
Deductibles can vary wildly depending on someone's insurance plan. However, there are a myriad of financing options, charities, etc if for some reason you can't afford that.
I would estimate the vast majority of people in the US need to budget at least $5k for a baby, if everything goes right. It would be better to have $10k cash though depending on prenatal visits, c section possibility, and multitude of other costs that can arise.
Do most people just decide to do it at home and take the risk?