Lots of these meanings seem to be cross-related.
Stave (also staff) meaning a stick, or walking (fighting) stick, could quickly progress to meaning 'moved quickly' (from someone walking quickly with aid of a stave)
Also to stave in a barrel, presumably need to hit it firmly, with the force concentrated at one point, easier if using a firm walking pole.
"Stave off" (hunger etc.) metaphorical?, keeping something 'at bay' as if by use of a stick?
Staff/stave, lines on paper, looking like 5 sticks, one above the other.

Could 'stave' (in its many forms) be a back-formation of a singular noun? Plural of 'staff' is 'staves'(Like leaf/leaves), singular of staves might then be stave.