1. Is it worth (money and time) signing up for one of these weekend trips with a Singapore dive shop OR should I make my own way to Malaysia and do the course with a local dive shop?
It depends on whether you have more time to spare for a longer trip. Travelling by yourself means you have to coordinate the bus trip and the boat trip to the Malaysian islands. If you take a bus leaving to and from Mersing (closest to Tioman), then you would have to spend a few hours waiting for either the ferry or the bus. You have to know where to alight on Tioman.
The dive shops usually get the ferry times and their private bus charter services lined up so that there is not too much waiting time needed. Less waiting time means more rest time and more diving time.
Since you need 5 dives to complete your AOW, you should check if you have enough time to do so. If you can dive on weekdays instead of the weekends, there would be less divers in the water.
2. And for either option, Singapore or Malaysia, does anyone have recommendations for which dive shop/operator to go with?
I went with Eko Divers. They are part of a larger group of dive schools bought over by Gill Divers -- as far as I know, Amazing Dive and Deep Blue Scuba are also part of it. These dive shops/schools pool their students, instructors and divemasters together and set off for Tioman weekly in a group of 60-80 students. That's how they keep costs low.
A class might have 5-7 students, 1 instructor and 1 divemaster/assistant instructor.
3. Which dive spot is better from Tioman, Dayang, and Bintan? Or are there other recommended spots in Malaysia close to Singapore?
I have not been to Bintan.
AFAIK, from March to October/November, Tioman is THE destination of choice for OW, AOW and Rescue diver student training. Then the monsoon season sets in, and then the training grounds become Batam, Bintan and Pulau Hantu in Singapore.
There is definitely a lot of things to see in Tioman. I hear Batam and Bintan suffer from poor visibility. Pulau Hantu has quite poor visibility.
Price-wise, the recreational use of the waters around islands of Dayang and Aur are taxed by the Sultan of Johor, so they are more expensive. I hear it is about 150 ringgit per day. But with less divers going there, I think the marine life might be richer.