Bali is really cool but it is wildly touristic/digital nomad now. Kinda depends on what you're into but further west and north from Kuta will be more of the party stuff in places like Canggu or Seminyak, and up north in Ubud and even farther its much quieter and relaxing. Ubud has a ton of wellness retreats and hippies. Try to do the sunrise volcano hike (Mount Batur, accessible via Ubud). Lots of really beautiful nature and waterfalls near there too.
Plenty of beautiful beaches. Beaches on the west side will be filled with Australian tourists, beaches on the south near Uluwatu, like Pedang Pedang, will be a bit quieter, and beaches on the east side will be the least crowded. Gili islands are reachable via ferry and so is Nusa Penida, the latter is quieter than the former.
Eat Babi Guling while you're there. It's like suckling pig, but it is a greasy gut bomb, so make sure you have a good toilet at your hotel. Check out some temples for sure. Uluwatu temple is really famous in part because of the incredible cliff-side view over the ocean.
For your other question, you can totally knock out Singapore on either side of your trip. But if it were me, I'm hopping on a cheap flight to Yogyakarta and checking out Borabadur and Prambanan. Borabadur is one of the coolest things I have ever seen.
