Out he rushed,with a stick in his hand.什么时候用Out did he rush.

只能说Out he rushed,with a stick in his hand.他手中拿着棍子冲了出去
Out,away,in,up ,here,there 等表方向,位置的副词置于句首时,如果主语是名词,要用全部倒装语序
如：
Away rushed the children.
但如果主语是代词,不能倒装
如：
Here he comes.他来了.
比较：
Here comes the bus.汽车来了.
你说的“did he rush.”是部分倒装,这里不存在这种情况.