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If by old flowers and buds you mean those that are brown and dead, certainly you can (should) cut them off. This article will tell you more about pruning your rose of sharon: https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/pruning-rose-of-sharon.htm

Unfortunately, about the only way to stop this is to keep the plant from going to seed by cutting off the faded blooms and discarding them. You should also try to keep the spent blooms picked up from around the shrub/tree. As for the volunteer seedlings, you'll have to keep them pulled. If the Rose-of-Sharon is located in the yard where it can be mowed around, this will normally be enough to keep the young plants from popping up. Otherwise, you will either need to hand pull them or you can try placing mulch in garden areas to limit their growth. This article will help with dealing with volunteer weed trees: https://www.gardeningknowhow.com/ornamental/trees/tgen/weed-trees.htm

If by "starter" you are referring to rooting of cuttings, this article tells you the basics: http://www.ces.ncsu.edu/hil/hil-8702.html
It may also be possible to dig up offshoots from the roots that carry some roots with them, and plant those.

The winter hit your poor plant pretty hard, but there's one strong branch left, so with time, care, and pruning, you should be able to have your plant looking beautiful again in a few years. Here's information on pruning: https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/pruning-rose-of-sharon.htm
and general care: https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/rose-of-sharon-bush.htm

If you dig around the old dead trunk to make sure you're not cutting into any live roots or pieces of trunk, you should be able to saw the dead trunk out without any problem. Here's an article on pruning rose of sharon: https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/pruning-rose-of-sharon.htm

Plants often lose some leaves because of the stress of changed environment and soil moisture. Make sure to water your plant well every couple of days for the first 2 or 3 weeks, then every week for the next 2 or 3 months. More tips here: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm