The arrival and departures areas at HKG are segregated, so upon arrival, if you have your BP for SQ865, you just need to pass through security (no need to go through immigration) to get to the departure areas and the lounges. If SFO couldn't print the BP for SQ865, go to the transfer desk to get the BP then proceed through security.
I think SQ should be able to through-check your luggage, so that there will be no need to reclaim it and check in again.
In case you want to store hand luggage in the lounge because you don't want to schlepp it through your sightseeing trip, that is perfectly possible in the lounge. However, they don't have lockers, your luggage needs to sit in the open racks next to the entrance and the lounge ladies make it very clear that they do not accept any liability whatsoever for it. So you have to keep wearing your jewelry...
I'm not sure about the mechanics of going from the departures area back into the city. I haven't had an itinerary that made a quick visit to the city feasible.
In order to get from the departure area back into the arrivals area (and then onto through immigration into Hong Kong) you need to be accompanied by an airline employee. The SQ ladies will be happy to call somebody from the SQ ticket desk to take you through - it might take a couple of minutes, though.
Have safe travels and enjoy Hong Kong!!
Primenumber