"if the covered Person's insurance should expire, the policy will pay under this benefit providing conception occurred while the policy was in force." If this is indeed in your policy, they should pay, provided it can be concluded that conceptino date was in that duration - it can be deduced from your delivery date with reasonable accuracy.
如果拒付是不是要由保险公司给我个正式通知? Yes, they will notify you by writing for any claim. The hospital made the claim on your behalf.
是否我必须要续保险一直到小孩出生? Yes, you should, for visa status reason.
如果不续会有什么不利的后果? It could be small issue if no one investigates it. It could be big issue if for some other reason the INS wants to verify your legal status. When you let your insurance expire, you already violated your F-1 and F-2 visa status. It could be big issue in the future.