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You can, though it's sometimes difficult. This article will tell you how: https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm

This article should help: https://www.gardeningknowhow.com/ornamental/shrubs/forsythia/forsythia-pruning.htm

You can actually transplant the shrub now as long as the threat of frost has passed in your area. These articles will help with caring for forsythia and preventing shock when transplanting: https://www.gardeningknowhow.com/ornamental/shrubs/forsythia/forsythia-shrub-care.htm, https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

As long as you don't take more than a third off at one time, it's fine. This article may help, if you haven't read it yet: https://www.gardeningknowhow.com/ornamental/shrubs/forsythia/forsythia-pruning.htm

If you have not pruned any, it may be in need. Pruning can help rejuvenate the shrub and improve its overall health and blooming, as well as its appearance. You can do this now or wait unitl the few blooms you do have fade out. These articles will help: https://www.gardeningknowhow.com/ornamental/shrubs/forsythia/forsythia-pruning.htm, https://www.gardeningknowhow.com/ornamental/shrubs/forsythia/forsythia-not-blooming.htm In addition, if blooming remains an issue, it is most likely lacking phosphorus. Add some phosphorus, such as bone meal, to the soil.

You can transplant now only if you can ensure that the shrub will receive plenty of water while it re-establishes its root system. In your area, the summers can get pretty hot and this could have an impact on how well the roots establish. Therefore, you may want to wait until fall to transplant so the heat will not be too much for it.