1) If you're ordinarily resident in HK on either an employment visa, dependants visa or any of the long term derivatives like entertainer, education, internship, working holiday, you should qualify for a "HKID". At the end of the day, it is an identification document, not a status.
2) If you just need to register with a doctor, you can do so with a passport. This is used mostly to catalog your records... not to prove that you're here legally. Most doctors will be ok with you going to them and stating that you're visiting. However, if you want free / low cost govt services - then you do need to have a HKID, which implies that immigration has accepted that you have an appropriate visa.
3) If your child qualifies - do you have a spouse who is already a resident? Or was the child born here in HK while your spouse or you were resident here?
4) A deed poll implies that there is a name change involved? Few threads here which might give you some guidance - http://hongkong.geoexpat.com/forum/tags/deed%20poll/