Changi Transit Hotel

You need onward flight details to book a room in advance at the Tranist Hotel. Whether that means u need to show a boarding pass on arrival I don't know.
I stayed there without a boarding pass about 15 years ago, was just a walk up and check if they had any rooms left.
If u are arriving midnight or later I'd say you would need an advanced booking these days.
With regards to collecting your baggage, officially u can't but have heard of people asking the immigration officials if they can go thru and collect their bags as they are staying at the transit hotel and they have been allowed to do so. This may have changed post 9/11. Anyhow your left baggage is quite secure with sub-machine gun armed Gurkas guarding the airside secure areas.
You can email the hotel and ask whether u need a boarding pass to book a room here, just click on "enquiries" top right of page.

You might find it's more convenient at the end of the day just to stay in a place at Geylang, which is a short taxi ride from the airport. I don't know your budget, but any of the Hotel 81 chain would be cheaper than anything at the airport, even including the taxi ride.

since you've got so much time , if you manage to reach the train station by 11.45 , you should be able to catch the last train back to the city , if not a taxi ride would include at 50% surcharge after midnight and a 5$ airport surcharge if you took a cab to the city . and find a decent hostel to stay at.

unfortunately with a day and a half between flights I don't think this will work; to use the transit hotel you stay airside and don't go to baggage reclaim, so your luggage would be going round on the carousel until someone takes it away or blows it up. You've no option but to go through immigration.

book into a Geylang hotel as mentioned - I like the Fragrance Emerald, not flash but clean and quiet, they put the tourists on different floors from the 'short time' guests. As you are landing at midnight you will be too late for the MRT, so it will need to be a taxi. Next day the hotel will keep your luggage (Checkout midday) while you explore, a bus will get you into the centre of the city in 10 minutes. To return to the airport, take the MRT from Kallang which is about 10 minutes walk from the hotel with luggage and at a pace suited to the climate.

If u don't collect your luggage within 30 mins it is collected by the luggage attendants and taken to the "Lost and Found" counter where it is kept until collected.
So u can stay in transit and use the facilities, transit lounges or hotel and when ready, clear immigration, collect your baggage from Lost and Found, clear customs and check in for the HKG flight landside.
Unless one of your flight legs is with one of the budget airlines, u may be able to check your baggage right thru to HKG, thus solving any problem.
But if any one of the flights is with a budget airline u will not be able to book a room at the Transit Hotel and should go into Geylang for a onite room. Regin Hotel is a good choice close to Kallang MRT, single room from $65sin, http://www.reginhotel.com/standardsingle.asp

i would recommend lodging at le peranakan hotel which is nearer to change airport then geylang but is more accessible by cab.