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The ASPCA list's Eucalyptus toxic to dogs, cats and horses.

The dangerous component is the oil in the Eucalyptus plant.

It might be useful to speak with local experts, such as an extension agent at a local university.

For seed germination and planting, please see this guide to growing eucalyptus from seed:
http://www.angelfire.com/bc/eucalyptus/seed.html

Your species is not discussed specifically, so you should assume that its seeds need to be cold-stratified (chilled) before germination. This species may have poor germination rates, so plant many seeds to ensure that some germinate. Keep the seeds evenly moist until they germinate.

The sunlight indoors are likely not enough for the full sun plant. Adding a grow light will help the plant through the darker winter days.
You can cut back the plant and allow it to go dormant in a cool and dry location--basement would be fine.

https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/eucalyptus-houseplants.htm

Fertilizer is generally not needed for members of this genus. In some cases the Eucalyptus can become iron dificient in iron.
It can be helpful to apply chelated iron in the fall and early spring, for young developing trees.

https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/tips-growing-eucalyptus.htm
https://www.gardeningknowhow.com/plant-problems/environmental/leaf-chlorosis-and-iron.htm

You need a growing zone of USDA 10 or higher to support a Rainbow Eucalyptus.

https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/rainbow-eucalyptus-tree.htm

I think losing lower branches is normal for this type of tree. As long as the upper branches are healthy and the tree is still growing, everything should be fine.