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Rooting hormone is a substance you can buy that has fungicide and plant hormones. The fungicide helps to keep away rot and the plant hormones will encourage the cutting to grow roots. It is available at most nurseries, including those outside big box stores.

Soilless mix typically refers to potting soil. It means a planting substance that does not have soil, as in the stuff the ground is made of, in it. Instead, it is made of plant based material, like peat moss. Soilless mix does not compact down like soil does, which is why it is good for rooting cuttings.

This article will help: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

It is not necessary to cut either of these shrubs back when transplanting. In fact, the extra foliage may offer additional winter protection if transplanting in the fall. Here is an article or two on pruning each of these shrubs that you may find helpful: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm
https://www.gardeningknowhow.com/ornamental/shrubs/azalea/pruning-azaleas.htm

I would wait until spring. Pruning now may cause the bush to start growing again instead of going into dormancy which would leave it vulnerable to the winter cold. Instead, I would try wrapping the lower trunk with tin foil. This will keep the rabbits away from the bark until spring.

This article will help you with that: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

There are two things that can keep a burning bush from turning colors. One is too little sun (aka too much shade) and the other is the temperature does not get cool enough. I would imagine that where you are it gets cold, so it might be a light issue. Is the plant located in a shady, or even somewhat shady location?

The other possibility is that it was a mislabled shrub. You may have a green euonymus, rather than a red one. They are difficult to tell apart and sometimes the growers can get them mixed up.