“Does she do the work?” is the right sentence.
If you get confused apply all the available options one by one. Soon you will realise and figure out all the wrong options.
Now let’s observe the aforementioned options
It is crystal cleared that Does and Doing are the wrong options.
Now, does she do the work? and does she not the work.
As there is THE WORK which indicates the importance of the work, therefore, “do” is the right option. Which is used in order to do focus on something or give something importance.
In case if there was only Work, then
Does she not work? seems correct too.
Because there is focused on the girl(she) here.