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You can. It may reduce its blooming for next year, but will not harm the plant and the blooming will recover for the year after that.

This would be ok. We would recommend that you wait until early fall to do this.

Yes, I would remove the dead branches. For more info on pruning these trees, this article will help: https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/how-prune-pussy-willows.htm

It will most likely come back, but you will probably not have catkins next year. We would recommend mulching it for the winter, just in case and to help protect the tender new growth that will happen due to the late pruning.

Yes, you can do this in fall or spring, whichever you prefer depending on your climate. You will want to dig up as much of the rootball as possible, and when you move it to the new location, cut back the plant by about 1/3. Make sure to give it plenty of water and a phosphorus rich fertilizer after transplanting. This article should help reduce transplant shock: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

This article should help with that: https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/how-prune-pussy-willows.htm