You've done a good job investigating the roots and trying to address the problem. Penn State Extension has a very clear table summary of common problems affecting your plant.
I think your attention to your roots is a good first step in addressing the problem. The Penn State Extension, below, points to the brown roots as evidence of root rot:
I personally would not jump to applying fungicide. Obviously, you are interested in saving your plant rather than discarding it so don't do that either :)
This article (under Method 2 for indoor plants) details a few basic steps to take in addressing root rot. You've already done step 1 in looking at your roots! While your roots are not black and the worst they could be, it does look like almost all are mushy.
You have one root that looks much healthier comparative to the others. It is the whitest and appears the most spry/rigid root in the bottom center that has a few small branching roots. In order to reach your goal of having all your roots look like that one little root, I would cut off the most mushy and darkest brown parts of the roots.
Because you only have the one little white root, I would not cut off all the brown roots as you would be left with only one small root, but aim to thin especially the roots nearest that healthy root to hopefully prevent spread.
As the article details, I would then try to remove the leaves in equal proportion to the roots you removed. Unfortunately, as you said, your poor plant has already lost leaves in stress, so I would take care not to rid it too much of its leaves. The tension of course is that the root rot could spread. As almost all the roots are so mushy, it may be too late to address the problem. BUT! Try these steps as it doesn't take much effort and plants can be remarkably resilient.
I would make sure when you do eventually water it, to take care to water at the base rather than from the top over the leaves- that may already have been your plan.
Was this plant in a pot when it was outdoors? As the problem was most noticeable when it moved to your house, I would try to address the factors that were likely most affected. I expect a brighter location, but not in direct sun unless in the morning or late afternoon, would positively affect your plant. After awhile, if your roots look better, I would try a higher humidity location like the bathroom per the Spruce's guide on best conditions for Cordyline plants.
I would also be concerned with the white fuzzy material on your plants stem- I would start by trying to wipe it away with a damp cloth.
As to your other questions- I do not know about the spots on your leaves. I imagine they may be related to the light it is receiving or simply symptoms of the stress. I personally would address the roots, the white fuzzies and growing conditions and not worry at all about the little spots. It may be addressing the above factors will prevent the spots on future leaves. I also do not know about cutting the stem of these types of plants. Personally, I would wait to do that until after I had addressed the other problems as the plant has already dealt with a lot of stress and that would add significantly to its many "battle" fronts.
Best of luck, I hope this answer gives you some starting points.
