1) You need a visa for China, no visa for HK. You only need a single-entry visa for this itinerary. If you need a multiple entry for future visits, you're best bet is HK not anywhere in the USA. Even UK is difficult to get multi-entry for UK passport holders.
2) With a greencard, you can get a single-entry visa in the USA on your British passport. Will probably need to show your flight reservations and hotel reservations in Shanghai as part of the application.
3) Getting a Chinese visa in HK is absolutely doable, but in order not to waste time, use suggestion above and go through Forever Bright agency (reasonable fee) to do the legwork and free up your time, instead of queueing at the Commissioners Office in Wanchai. FB can do next-day service. Processing generally not done on weekends or Chinese holidays.

If you're going for a tourist L visa, no invitation letter is needed. If going for a business F visa, invitation letter needed in NY, often not needed for a Brit applying in HK if you use an agent, but best to check FB to see what the latest is.