Questions about the meaning and position of 一下
Question 1. Does the 一下 mean how many the speaker smacked her/his child, only one time in this case, or act as an adverb like 'a little bit'?

It means only once here.
Note that 一下 might mean a little bit too, depends on the context.

附於動詞後面，有略微的意思。如：「想一下」、「看一下」。亦作「一下子」。

=&gt; Attached after the verb, means a little. (Note: In some instances, depends on the context. --end note)
The other two common meaning of 一下 is in a short while and suddenly.

(1) [in a short while]∶表示短暂的时间
灯一下又亮了
(2) [in one go;at a single stroke;all of a sudden]∶突然间
天一下阴了下来

then

Question 2. would there be any difference if I change the place of 一下 as follows?
你去问一下老师 → 你去问老师一下
打孩子一下 → 打一下孩子

No, there're no difference. In both cases 一下 acts as an adverb modifying the verb, the position won't change the basic meaning. (Except that the object will be emphasized if it's placed at the front.)