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I would say yes. Since Rose of Sharon shrubs are actually a variety of hardy hibiscus, cutting would be taken much the same way. They also self-seed when not deadheaded, so you just might find some seedling growing beneath the shrub, which you can easily pull and pot up or transplant elsewhere. These articles will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm
https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/rose-of-sharon-bush.htm

Rose of Sharon is a variety of hardy hibiscus, and can be easily propagated through cuttings. This article should help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/hibiscus-propagation.htm

In addition, the plant self-seeds readily when blooms have not been cut, so you may find young seedling growing beneath the shrub as well. These can be lifted and grown in a container or transplanted elsewhere.

These articles will help you with hydrangea and rose of sharon:
https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/prune-hydrangea-bushes-hydrangea-pruning-instructions.htm

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

The mountain laurel should be pruned in late spring.

Rose of Sharon need no additional care for winter. If they are new plantings, you can mulch around their base to ensure that they are fully established.

This article will help with that: https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/pruning-rose-of-sharon.htm

Rose of Sharon is a hardy plant and should be fine with no extra care. But since they are new plantings, if you want to mulch around the base of the plant for added protection, that will not hurt them and will help protect them if they have not fully established yet.