I think you are just thinking too hard to be honest. You know that there is no slip on the surfaces. Therefore, you know the fluid velocity at each of the surfaces and you know that it just varies linearly from one surface to the next.
It should be straightforward from that point to answer part A since you know all the dimensions on the problem and the necessary velocities. The fluid is divided into two sections (above and below the plate) and you have enough information to get the velocity profile in both sections with minimal work. You can use those profiles to easily solve for the heights where the fluid velocity is zero (there are two such points)
For part B, you just use the equation for shear stress:
[tex]\tau = -\mu \frac{\partial u}{\partial y}[/tex]
That will give you a stress (you need one for the top and bottom of the plate) that you can multiply by the plate area to get force.
