From the experience Yuantongsi noted above of his friends in similar situations, each country would seem to recognize citizenship of its own country, but not the other. But that would be ok, be Chinese in China and be American in America, best of both worlds!
I would say baby bed is not absolutely necessary. There is not even united opinion about that in the West, but probably overall it is preferred. Of course by Chinese it is not preferred. Not a big deal really, whichever you want - unless you think there is a risk of a parent rolling on top of the baby! But after a couple of months if the baby can go to a crib, then it means the parents can get a little bit more sleep - which helps everybody.
You say there is no sunlight in your house? Having sunlight would be an advantage I think, especially if you don't leave the house at all. Check with a doctor you trust. (There is a really good American paedatrician in town.) If you don't want to move house, then perhaps tradition could bend for the sake of the baby's health, to take him or her for a walk sometimes in a stroller with a shadecloth.

Wow, you guys really love China so much to marry into a Chinese family! :)

Great news about the de facto dual citizenship!
Hope all goes well for the birth.
You should be aware that the normal situation is that in most Chinese hospitals (especially with a Chinese mother), the father is not permitted to be present during the birth. If that is important to you, you could try to shop around & negotiate for that, saying that you have a different culture and it's a vital part of the birthing process for you to be there etc. I haven't heard whether anyone has had success with that.

Glad to see there is some specific relevant info there provided by Mr Geezer. Maybe the registration doesn't have to be within the first month. And don't be daunted by the paperwork required by the embassy. It will all be worth it for your child's future.
As for schooling, just like anything with having kids, it all costs money. Like their health for example, we can't afford to count the pennies too much, we have to pay what we need to pay to make sure they have good medical treatment when required. Same for their education. That being said, there are a few private schooling options around town that would vary in cost, some hopefully within your budget.
Most importantly, don't forget if you want your child to enter the American education system in the future, they need to be getting at least some of their schooling in English, so that the transition will be manageable. But that doesn't mean you should overlook the advantages of some Mandarin instruction either. So I would suggest your best option is to look into the bi-lingual school situation, as there are some around.

Sometimes the consulate replies to their emails too.
From my observation of Chinese weddings, I have seen the separation of the 'legal' and 'social' elements. Chinese people marry legally at the Registry office, but often they or their families do not consider themselves really married, or married in the eyes of society, until they hold a celebration, like what we call a wedding reception. Of course in Australia it's usually both together, but it doesn't have to be. So if there was any procedural benefit, you could hold the legal part in China and the social part in Oz. It seems some Chinese people treat the 'legal' marriage like what we call engagement, and perhaps do not even live together as man & wife until the 'social' wedding.
But if the procedural requirements are just as strict, then you can only do things like search the websites and keep trying the consulate or even the Beijing embassy.