How do you care for a Japanese Weeping Willow Tree?

You need to know why it is dying and how far gone is it. If it is terminal cut it down and find out what killed it before replanting.

Only when you need to and remove the required branches in winter

There is no state with the weeping willow as its state tree.

Like aspirin. They have an astringent sharp smell. Last time I checked, trees can't smell.