Where should I plant a rose of sharon and what time of year?
This article will help you: https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/rose-of-sharon-bush.htm
Should the dead seed pods be cut off, and if so, when? How do you tell a male holly from a female?
You can cut the pods off at any time. It will not harm the plant. Normal pruning is done in in early spring.
This article will help with male and female hollies:
There are tiny black dots on the plant that I assume are bugs, because I have another plant that does not have it.
They are likely bugs. Many of the smaller bugs attach to the plant and move very little. Treat the plant with neem oil and that should take care of them.
I have about 40 Rose of Sharons that I dug up with roots. How do I store them so I can plant them next spring?
I would personally pot them very tightly in a few containers and keep them like that until next spring. Keep them watered and they should be fine. You can separate them back out in the spring to pot individually or plant in the ground.
Home growers don't have the same access to chemicals that keep bare roots hydrated for extended periods of time like commercial producers do. Even then, commercial producers will only keep plants bare root for a season. The plants would not be able to survive from now until spring base rooted, so just potting them up tightly and then keeping them that will will work.
Can the roots of a rose of sharon shrub do damage to basement cinder block walls? I have 2 large ones planted next to house that have been there for years, and somebody said I should remove them, but I didn't think roots of a shrub could damage basement walls.
The root system should not harm anything. Be aware though that these shrubs can be a bit invasive in certain climates, and if this is the case in your climate, you will need to be diligent about cleaning up seedlings to keep them from over running the area.
We have 6 Rose of Sharon plants in our yard here in Garner, N. C. They have lavender double blooms. A couple of years ago, there appeared on one of the branches a pure white bloom. Both colors on the same branch, only a couple of inches apart. In the twenty or so years that we have had these plants, nothing like this has ever occured, either before or since. The question is: just how rare is this? Since it has never happened before, I assume it must be pretty unusual.
It is indeed rare and something quite special to behold. It is a genetic mutation thing that may not ever happen again. Or it could happen again next year. Once it has happened there is always a possibility that whatever caused the mutation will come around again. I would have the camera ready for sure, I hope you did the first time it did this. Also, this can be done by grafting but then you would have a whole branch or series of branches with different bloom colors from the main one.
I have a large Rose of Sharon tree. Another tree started from this one and I would like more to start. What makes a new tree or bush start?
Most of the time, these trees will drop spent blooms, which left on the ground will end up reseeding and producing more plants. That's why many of them have little sprouts coming up around the trees. While the little sprouts can be dug up and replanted (keeping roots intact), you can save the spent blooms for seed harvesting to make more or start some through cuttings as well. They are propagated the same as other hibiscus plants, so this article will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm