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The first thing I'd do is treat the bush with insecticidal soap a few times. Here's some info on that: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/homemade-soap-spray.htm.

Then, I'd introduce some beneficial insects to get the problem under control. Ladybugs work really well on aphids. https://www.gardeningknowhow.com/search?q=beneficial%20insects You should be able to get these either through your local garden center or order them online.
Make sure to keep any dead leaves, blossoms and debris cleaned away from the plant. Early next spring try pruning it a bit so that the air can circulate throughout the branches.

Yes. Though your shrub's flowers appear white, they are actually what is known as "bicolored". This just means that there is still a shade of pink within the bloom. Sometimes, this shows more than other times.

This is going to be a normal occurrence, and there will be nothing to be done.

Here is an article that will help you with some tips:

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

Sprinkle the bone meal or fertilizer and then water the soil. I usually only apply the bone meal in Spring or when planting in the Fall in areas that really need phosphorus though. If the flower buds are falling down prematurely, it may be due to lack of water or a nutrient deficiency in the soil. Some sandy soils can have nutrient deficiencies. If the RoS is not developing flower buds, maybe there is dense shade or too much nitrogen in the soil. If there is too much nitrogen, reduce high nitrogen fertilizer applications and only apply per label directions. There are some soil kits sold at garden centers that can help you determine if your soil has too much nitrogen. Insects can also eat away the flower buds. Japanese beetles can also cause a lack of blooms by eating the flowers en masse.

The shape depends on how it is pruned! Cultivar and care will dictate height.

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

As far as where to buy, we do not offer that information. I would check with your local nurseries or online nurseries.

Some choices are to use pre-emergents chemical herbicide products like Preen, plant a non-seeding variety (to avoid this problem completely in the future), deadhead the spent flowers (now and every year), pull them by hand (sometimes I prune the bulk of the small plants and they do not return), cover them with cardboard for a few months or very carefully paint them with a regular herbicide for woody plants like Bioadvance's Brush Killer or Ortho's Ground Clear.

Pictures of the damage can sometimes suggest the type of insect: leaf cutter bees, weevils or even Japanese beetles. You need to catch the culprit in the act -some do the damage only at night- in order to apply appropriate counter measures. You can also take samples to a local plant nursery or garden center for identification; put them in a sealed, transparent plastic envelope. Include a few leaves that will show the type of leaf damage.