How to root tropical hibiscus.
This article will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm
I recently transplanted my hibiscus plant from this summer. I've never had one before and it is quite large in the house. Some of the leaves are turning yellow and falling off.
As long as it is in the house, you can cut it back by up to 1/3 before bringing it indoors. Otherwise, you should wait till spring. This article will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/tips-for-pruning-hibiscus-plants-when-to-prune-hibiscus.htm
The yellowing and leaf drop signal that the plant is getting ready for dormancy and is normal. This article should also help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/wintering-hibiscus-indoors-winter-care-for-hibiscus.htm
There are several reasons this can happen, normally associated with stress of some kind, or it could simply be going dormant. This article explains many of the causes 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 stems have little bumps on them and I found a colorful bug on it, not sure if the found bug is the cause. I live in tropical environment.
The bumps are scale insects, which neem oil should help with, regardless of the insect pest (as there could be another culprit feeding on the leaves). Here is more iinformation: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm
https://www.gardeningknowhow.com/plant-problems/pests/scale-bug-how-to-control-plant-scale.htm
I don't want to dig up my hibiscus tree to bring inside for the winter. How can I protect it through winter and leave it planted outside? It is a hibiscus rosa-sinensis.
If it gets colder than 20F where you live, this plant cannot stay outside. It will not tolerate temperatures under 20F and there is no way to protect it.
I live in zone 5 and keep hibiscus outside in winter with mulch. Am I supposed to cut them down in fall?
No. It is better to wait for spring to do any cutting on this plant. Cutting now will likely lead to cold damage this winter. Here is more info: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/tips-for-pruning-hibiscus-plants-when-to-prune-hibiscus.htm
What can cause recently planted Hibiscus green leaves to turn lemon yellow?
Most likely the stress related to being newly planted is causing this. It is nothing to worry about right now, but just in case, here are some other reasons the leaves may be turning yellow: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/answers-to-what-causes-hibiscus-leaves-turning-yellow.htm