It is likely a tropical hibiscus, Hibiscus rosa sinensis.

I have had similar experiences with bringing them in and having them flower into January. I believe the flowering stops then due to the lower light levels indoors. I put mine in my best sunny window. I even tried buying a heavy duty plant light. Nothing made much of a difference. I think that time of year you just have to focus on keeping the foliage healthy and pest-free and wait until summer for blooms.

Any time you prune a branch it takes up to three months for that branch to bloom again. If you prune it back when you move it indoors, you will be depriving yourself of flowers that you could have until January. Of course, if it's too big to bring indoors, you may just have to prune somewhat.

One good trick is to prune 1/3 of the branches at a time, either scattered throughout the plant or on one side of the plant. A few weeks later, prune 1/3 more, then in a few weeks do the last 1/3. This keeps the plant always making blooms and keeps the size manageable and the shape nice.

You'll get occasional yellow leaves, not a big deal unless you have lots of them. It generally means something has changed from the status quo, less or more light or water, higher or lower temperatures, etc.