Is it possible to use root stimulator to start new hibiscus cuttings?
You can use a rooting hormone, if you would like, though you don't necessarily have to. This article should help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm
What can I do so my hibiscus blooms? Today I cut many branches, and I put some of them in water, so I can get more. Is this okay?
Use a fertilizer higher in phosphorus to encourage more blooms. If you're looking to produce more plants, then this article will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm
The leaves are suddenly turning yellow. Bush is several years old and appears to healthy up to this time. Blooming beautifully.
There are several reasons this can happen. As long as the plant is healthy looking otherwise, it should be fine. This article explains many of the reasons for leaf yellowing and will help you decide which is likely for your plant: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/answers-to-what-causes-hibiscus-leaves-turning-yellow.htm
The bud appears to be cut off clean.
A number of things can cause this to occur. This article should help you with that: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-flowers-hibiscus-blossoms-falling-off-plant.htm
Our Hibiscus plants look dead. We think they are dead except when you break a stem, it looks green. Makes you think there is life in there somewhere. How can we tell if they are dead? If they are not dead, what can we do to bring them back to a green and blooming plant? Many thanks for any help. . . Shirley
It may just be slow to start. If the stem is still alive, it will regrow. This article will help you determine if they it's still alive: https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm
Should hibiscus plants be dug up and separated after they have reached a certain width? And is it okay to move them inside after already being outside for a couple years?
No they do not need to be divided, but they may require pruning to keep them at a manageable size. This article will help with that: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/tips-for-pruning-hibiscus-plants-when-to-prune-hibiscus.htm
If they have been growing in ground for a number of years without any issue, I would let them be. Digging them up and putting them in a pot to bring indoors may only put them into shock. However, if you would like to start a new plant and grow this indoors, that would be a good alternative. This article will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm