I am almost definitely sure that your helper's child will have no residency rights in Hong Kong UNLESS the child's father is a resident--then there is a possibility. Is her husband either a temporary or permanent resident in Hong Kong? If he is a temporary resident then he can apply for the child to have a visa here. If he is a permanent resident, he can apply for the child to have permanent residency. Otherwise, foreign domestic helpers have literally no way to become residents in this city--a friend who had a long-term Filipina girlfriend told me that even if he married her she still would not be able to get residency because she had come here as a FDH. You need to research this and find out for sure. The HK Gov. is not stupid and they have purposely designed the system to prevent FDH from coming here, getting pregnant and then claiming residency for their children--it's a social welfare nightmare for them in that case.
About the car and all the other ridiculous demands--and the fact that she became pregnant after only working for you for a couple of months....wow. Let me say, that the whole thing that she got pregnant so soon after starting to work for you won't look good for her at all in front of any labour tribunal. I'll tell you why...generally, the rules are that there is a "blackout" period of about 10 months when you sign a new contract with a company--depending on your company. I work for the government in an indirect way and because I happened to give birth to my child when I was only 7 months into a new contract, although I was guaranteed maternity leave (they can't deny you that 10 weeks, by law) I had to take UNPAID maternity leave. This is because I was only 3 months shy of having worked for the company for 10 months when I gave birth. Basically, no one in HK looks favorably upon a woman getting hired in a job and then immediately getting pregnant or worse still (which was my case) getting hired when she is already pregnant. I just happen to have very lenient and understanding bosses--most HK bosses aren't. It's going to look like a total scam on the part of the helper if she takes it before a court. Believe me.