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Pruning during active growth will result in a tree bleeding sap.
Suckers should be pruned from the tree as soon as they appear.
Watch for signs of stress, pests and disease in the tree. Treat any pests with Neem Oil.
This link will refresh you on proper pruning.

https://www.gardeningknowhow.com/ornamental/trees/willow/how-to-prune-willow-trees.htm

Ah... the corkscrew willow! One of my favorites... It does contain salicylic acid, which is related to aspirin. The cat may dig in the pot, as they typically do, but this poses no threat to the animal. It is so bitter that the cat won't want to eat it after one taste. Carry on, and enjoy the awesome tree!

This might be a Salix matsudana, a type of willow. Varieties of this species with especially twisted branches are known as Corkscrew Willow. Here is more about its requirements:

http://www.public.asu.edu/~camartin/plants/Plant%20html%20files/salixmatsudana.html

Definitely keep it watered, as willows tend to like being, constantly moist.

Now, I would not fertilize it very often at all. Adding dolomite lime will help keep the areas pH buffered, but adding that much fertilizer will likely cause the leaf drop to get worse. I would have to guess that it just needs as much water as possible. Keep the area moist.

Here is something that will help you:http://aces.nmsu.edu/ces/yard/2008/080208.html

This sounds like bacterial wetwood, aka slime flux or alcohol flux. Willows are prone to this disease. Bacteria produce gas which makes the sap foamy and often foul-smelling. Check with your state extension service for recommended treatment in your area. Here is what New Mexico extension has to say on the topic. http://aces.nmsu.edu/ces/yard/2000/081200.html

Doing so would bring in a multitude of problems, such as pest and fungal infections. It would not be wise to do so. If you still plan to use this, you will want to water with 1 part peroxide to 3 parts water often to fight off any infection. Really, though, I do not recommend doing this.