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Bud drop is a very common problem with rose of sharon and other hibiscus species. This means the flower buds will turn brown and fall off before they open. Some level of bud drop is normal, but if all the buds are failing to open it could be that the plant is stressed by watering issues, heat, insects or disease.

These articles have more:

https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-flowers-hibiscus-blossoms-falling-off-plant.htm
https://hort.purdue.edu/ext/roseofsharon.html

Yes, you can promote additional root formation and also allow the plant to become more established in a pot before planting. See these articles for how:

https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/rose-of-sharon-plant-cuttings.htm
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm

There are over 200 different Rose Of Sharon cultivators. Many varieties are similar in colors and shades of colors. Without a specific variety it would be difficult to match a plant.
If your unhappy with the purchase I would contact the seller to see if the matter can be resolved.
Perhaps planting the new Rose Of Sharon in a different location is possible or gift to a garden friend.

https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/rose-of-sharon-bush.htm

One possibility is that this branch is a "sport", a genetic mutant which occurs on part of a plant, changing its coloration or other characteristics:
https://www.gardeningknowhow.com/garden-how-to/info/plant-sport-mutations.htm

Another possibility is that this is caused by environmental differences. The pigments in hibiscus flowers can actually fade in excessive heat or bright sunlight. If the different branch is exposed to more direct sunlight each day, this could be the cause.

You could either leave the branch on or prune it off, either will be fine. If you want to see whether it's a "sport", you could try taking a cutting.

I'd say it's possible to do this successfully. The reason to delay is to wait for the leaves to drop, but a transplant should still be successful if done earlier. If possible, choose a cool day on the humid side to transplant them. Water well the day before or the morning of the move.

Here is some more advice:

https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm
http://www.clemson.edu/extension/hgic/plants/landscape/shrubs/hgic1055.html
http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/planting-and-transplanting-trees-and-shrubs/

If there are no bugs on the plants, the most likely explanations for yellow leaves are excessive water or a lack of nutrients in the soil. See this article for advice:
https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/yellowing-rose-sharon-leaves.htm