You might want to wait until the tree starts leafing out before cutting. This way you can strip the bark right off.
You can get 4 staves, providing it's heart is centered. If it's heart isn't then take a stave from the small and bigger ring sides. Staves taken from the sides will warp sideways when the tree's heart is not centered.
Sure, you can go ahead and split hickory out into staves. Seal only the ends and store out of the sun and wind.
You can rough a hickory bow out close to bow dimensions for quicker drying. After a month or so in the proper conditions you can finish 'er out......Art