1. Yes DL/NW will send your luggage to Bali for you.
2. Why would you want to exit airside at SIN to retrieve the bags only to recheck them to DPS? You are better off checking them thru. If you re check them in SIN on a Y ticket you will pay excess baggage fees if the weight of the bag exceeds 20 Kilos.
3. Book the transit room well in advance of your transit date. Otherwise you may need to wait several hours for a room to open up. Another option is to exit the airside and check in at the new Crown Plaza next to the new T3.