#2
Hold it in place as you wind. If it still pops out you need to re-slot the nut. This would likely happen if you were replacing the original strings with larger ones.
#3
It’s hard to tell because the picture is so small, but it looks like you might be winding in the wrong direction.
#4
^^both of those things. And yeah, it does look like your wrap is going around the outside of the peg. You want it on the inside.