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There are two things that can keep a burning bush from turning color. One is too little sun and the other is the temperature does not get cool enough. Leaves are most brilliant in fall when there is a steady cold period. If you have been having warm snaps, these warm temps break down the chemicals that cause the leaves to be red, resulting in less brilliant leaves in the fall.

The other possibility is that it was a mislabeled 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.

I take it you feel the dog's peeing on the one bush is the reason it is much smaller than the other one. While that may be a possibility, there are others, and you need to investigate these possibilities also. One very likely cause is that the smaller plant gets less light, or the soil or drainage in its spot is less beneficial. It might even be in a less protected spot, and thus suffer more from winter weather. Also, it smaller plant may have some disease or pest affecting it; you might take some leaf, branch, root, and soil samples to the nearest county extension office for their analysis. This link will help you find the nearest office:
https://www.gardeningknowhow.com/garden-how-to/info/what-is-extension-service.htm
This article might give you some more ideas: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm
There's not much you can do about dog urine; a lot of water will probably do more toward washing it away than anything else. Perhaps this article about steps you can take to keeping unwanted doggies away will be useful. https://www.gardeningknowhow.com/plant-problems/pests/animals/keep-dog-out-of-garden.htm

You should wait to prune until March, when the chance of frost is much less. This article should be helpful: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

It is recommended to prune burning bushes while it is dormant, either in late winter or very early spring. It sounds like your burning bush is coming out of dormancy so I would make haste on pruning.

For more information on pruning burning bushes, please visit the following link:

https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

Was this particular burning bush sickly looking in any way, different than the others?

My best guess is that this may be crown rot. I am going to recommend you read the following article on this topic, which also prescribes a course of treatment:
https://www.gardeningknowhow.com/plant-problems/disease/crown-rot-disease.htm

For more information on the care of burning bushes, please visit the following link:
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm

How much of the bark is removed around the trunk of your bushes? Is the trunk completely girdled (bark completely removed around the trunk)? If so, the prognosis probably isn't very good and more than likely all the growth above the girdled areas will eventually die. However, if you are up for a challenge, you could try applying a bridge graft to restore the flow of nutrients in the bushes - more information on bridge grafts can be found here:
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-fix-girdled-trees.htm

If you believe the bushes are salvageable then I would recommend installing a rabbit exclusion fence or cylinder to protect your bushes from additional damage by rabbits. You may also want to consider trapping the rabbits, if regulations permit. More information on rabbit prevention measures can be found at the following links:
http://www.extension.umn.edu/garden/yard-garden/wildlife/rabbits-and-trees-and-shrubs/
https://www.gardeningknowhow.com/ornamental/trees/tgen/rabbit-damage-to-trees.htm

In the meantime I would recommend giving the bushes some fertilizer. For more information on the care of burning bushes, please visit the following link:
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm