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How long are your branches? A part of me wonders if this is just the normal growth pattern of an anacacho orchid tree. For your reference - here is an example of a young anacacho orchid tree with leaves clustered near the ends of the branches:
http://1.bp.blogspot.com/-tJZqizYTjhg/TZ-TPmVzgaI/AAAAAAAABxE/XdfXRg1v78A/s640/Anaconcho+Orchid_Apr2011.jpg

For more information on the care of an anacacho orchid tree, please visit the following link:
https://www.gardeningknowhow.com/ornamental/trees/orchid-tree/growing-orchid-trees.htm

The tree will probably start growing back soon after you trim it this fall. Anacacho Orchid Tree is a nitrogen-fixing plant (meaning it has bacteria in its roots that help it obtain nitrogen from the air), and it does not typically need fertilizer.

This fast growing tree should be blooming.
Check the growing conditions with the articles below.
Increased sunlight and a bit more water could help your tree.

https://www.gardeningknowhow.com/ornamental/trees/orchid-tree/growing-orchid-trees.htm
https://aggie-horticulture.tamu.edu/ornamentals/nativeshrubs/bauhinialunarioid.htm

Yes, they are part of the Orchids root system.
You can remove them if they are damaged or not healthy.
This article will help you.

https://www.gardeningknowhow.com/ornamental/flowers/orchids/roots-coming-from-plant.htm

If this is not fertilized at least once per year, then I would likely point this to be the problem. It would not hurt to fertilize this with a slow release all purpose fertilizer at least once per year.

Here is an article for more information about this species: https://www.gardeningknowhow.com/ornamental/trees/orchid-tree/growing-orchid-trees.htm

A Bauhinia could possibly grow indoors with conditions that you describe, although I've never seen it. I seriously doubt that it would flower well.