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As a former floral designer, I can tell you yes! You can use Fuchsia as a cut flower.

Make sure you use fresh stems and is possible choose flowers that are just about to open.
Soak your floral foam in water for at least 3 to 4 hours to ensure it is fully saturated.
Use a floral preservative in the water that you both soak the foam and when you fill the container.

When inserting the stems into the foam, push in the stem and slightly pull the stem back out of the foam. This will help the woody stems be able to take up water.

I would also recommend you locate a product called Crowning Glory. It is a clear liquid that you can spray on the entire floral arrangement when you have it completed. It seal the flower to prevent browning and wilt. This is a great product to ensure your hard work and pretty flowers don't suffer damage from delivery and warm weather.

This may help you.

https://www.gardeningknowhow.com/ornamental/flowers/fuchsia/fuchsia-plant-varieties.htm

I would recommend pruning in early spring or late August. I prefer spring, though. Here is a helpful article:https://www.gardeningknowhow.com/ornamental/flowers/fuchsia/pruning-fuchsia-plants.htm

It sounds like what you are describing are seed pods! You can harvest, and plant them when they are ready. Here is an article that will help you out: https://www.gardeningknowhow.com/ornamental/flowers/fuchsia/harvesting-fuchsia-seeds.htm

The best time to prune fuchsias is in the early spring, because this is thought to maximize flowering and prevent frost damage. However, it probably won't hurt your tree to prune it in fall as long as it is protected from cold. https://www.rhs.org.uk/advice/profile?PID=579

Yes, you can. This article explains how.
https://www.gardeningknowhow.com/ornamental/flowers/fuchsia/propagating-fuchsias.htm