PSB is the local police!
It’s required by law, and they will check it when you apply to renew your visa or apply for a new one. You bring your rental contract to the PSB and tell them where you live along with a photocopy of your passport and they give you a stamped document that you give to whomever is processing your visa.
Interesting that you mention the history of Hong Kong as a point of entry into China, because it was first visited and colonized by Portugese in 1513, exactly 500 years ago. I wonder if they observe that in Hong Kong.
I don’t think AMCHAM really qualifies as guanxi in China, they are not a diplomatic entity that represents the US government. It sounds more official than it is – they are an American Conservative lobbying group. For maximum guanxi points you want to befriend local government or police, people within the entry/exit bureau, etc. A lot of those guys are very accessible, even if you teach English to their kids – as long as they know you and you know them. The best situation though, of course, is to not get yourself into situations where you need guanxi!