To stay at the transit hotel, you must remain airside upon arrival in SIN. You will not have access to checked baggage. I've always used carry-on baggage, but there are lots of experiences posted here on FT that you can simply pick up your checked baggage after your transit hotel stay after you pass through SIN immigration and before exiting Customs.
I absolutely recommend staying that first night in SIN at the transit hotel. On my first trip US-TYO-SIN trip which arrived near midnight, I assumed I would want the transit hotel for 8-9 hours but actually was more than ready to go after about 5 hours, so I suggest booking the default 6 hours.
I travel for fun, and I like to stay at nice hotels. Arriving at a nice hotel after 1-2 am just seems not cost effective as I don't get the full value for that first day. That's why staying that first night in a transit hotel makes sense to me. A business traveler (spending other people's money) might have a different perspective.
I've stayed at all three transit hotels over the years, and they are all just fine and offer similar experiences. If you check a bag, you must enter SIN immigration/customs at the terminal you arrived because that's where your checked baggage is. Beyond that restriction, T1,T2 and T3 are all connected airside and you are free to visit any or all three terminals. (The new SIN T4 is completely separate and access-restricted).
Check out the Wiki at the top of this classic thread from the Mileage Run forum for ideas about SIN airport and Singapore in general:
https://www.flyertalk.com/forum/mile...i-4-9-cpm.html