Hi.

I arrive at SIN after midnight (EWR-TYO-SIN), so I was thinking of booking a transit hotel in Changi when I arrive for 6-12 hours (about $75 for 12 hrs). I would probably not stay the full 12 hours but it will give me a chance to crash when I arrive and awake rested to head into the city and a proper hotel the following night.

I arrive at Terminal 2 (ANA from TYO).

Questions about the SIN Transit hotels, which are in Terminals 1-3 (Aerotel in Terminal 1, Ambassador Hotel in T2 and T3):
1. Are they similar enough that I should just book in Terminal 2, or is one of the others significantly better and worth a weary traveler going to T1 or T3?
2. And if I do choose the hotel in a different terminal, do I have to clear customs first? (I think not).
3. If I check a bag, to avoid having to clear customs on arrival and re-enter, can I just leave my checked bag at baggage claim, sleep for 6 or 8 or 10 hours, and go to baggage claim in the morning to get it after checking out of the hotel? Is that a safe option?

Thanks in advance.