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Burning Bush is not "deer proof" and deer may munch on yours if there are hungry deer in your area. These articles have guides to deer-resistant plants:

https://www.gardeningknowhow.com/plant-problems/pests/deer/deer-resistant-plants.htm

http://pss.uvm.edu/ppp/articles/deerplants.html

Yes, fall is the ideal time to transplant a burning bush. Water the plant well the day before, and follow these guidelines to get the bush established with the lowest risk of problems:

https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

If your Burning Bush is slow to turn color or red it is generally related to environmental issue.
Dry conditions or even warm temperatures can delay the color changes.
Also is not unheard of for the variety of Burning Bush to be miss identified in the nursery or garden center.

In my own garden I have one shrub just now starting to turn as the temps have dropped.

Here is a link with care information.

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

Have patience and wait and see how the shrub does over time.

The burning bush is toxic to both dogs and cats so you may want to reconsider this choice.

Here is a comprehensive article on how to propagate a burning bush: https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-propagation.htm

Yes, that is a good plan. It will give the plant time to recover from the pruning by fall and grow more roots that it can take with it when it is transplanted.

Listed as toxic to dogs by the ASPCA.

You should consult a vet if your dog has ingested this plant.

http://ucanr.edu/sites/poisonous_safe_plants/Toxic_Plants_by_common_Name_659/