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If the tree has been girdled (where the bark is removed all the way around the trunk), it likely will not survive. You can try to do a bridge graft, but even those are not often successful. Here is more information about a bridge graft: http://aces.nmsu.edu/ces/yard/2007/090807.html If there is bark remaining on any side of the trunk, the tree has a chance of surviving. Treat the damaged area with a fungicide periodically to help prevent rot from setting in but do not seal the damaged area as this can trap disease in the wound. As long as there is bark that runs from the base of the trunk up, the tree has a good chance of surviving.

Have you pruned it? If you need to prune the tree, prune it now and then it will bloom for you next year. It only blooms on new growth, and when you prune, you cut off the new growth. If you prune now, you should have new growth again this summer and it should bloom off that next year. This article will provide additional info: https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/how-prune-pussy-willows.htm

This article will help you with rooting pussywillow cuttings: http://extension.unh.edu/resources/files/Resource000867_Rep913.pdf Once roots appear, they can be placed in pots or planted directly into the ground in spring.

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

Yes, pruning will not only keep their size more manageable but will also help when it comes to catkin production. This article will help with pruning your pussy willows: https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/how-prune-pussy-willows.htm

They prefer full sun but will tolerate some shade. While these plants like to be planted in areas with plenty of moisture, even poor draining, clay soil may pose problems with compaction. However, amending the soil can help with that: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm