Can I use the work done to calculate power of a motor? I am trying to calculate the energy/ power required to move a plate up and down in a liquid.
My approach is to calculate the forces that must be applied by the motor to push/pull the plate based on Newton's law (as a function of the velocity, weight of the plate, properties of the fluid, and so on). Then to calculate Work as the product between force and displacement. 
Can I say that the outcome is the required energy? Or is this a misleading assumption, based on the fact that both work and energy have the same measuring unit?
 A: Yes, this is basically the correct approach. Strictly speaking, some of the energy you put in will go into heating the water, i.e., the water will get hotter as you agitate it. The energy balance would be something like:
$$ \Delta E = W + C\Delta T $$
where $C$ is the specific heat and $\Delta T$ the temperature rise. However, under most conditions, the temperature rise will be small enough that it can be ignored.
A: Apparently (based on other answers) it depends on what your specific teacher(s) say(s)...
But yes, work and energy are the same thing, with only connotative differences; namely that work is the amount of energy by which a system changes:
$W=\Delta E$ (minus entropy losses if you're being picky)
I'd say it's like time vs. duration, if that helps.
If you have a known/required speed at which you need to move this piston thing, then the product of force and speed (or quotient of work and time) gives you power:
$P = \frac{\Delta Energy}{Time} = Force \frac{\Delta Distance}{Time} = Force \times Speed$
$P = \frac{\Delta Energy}{Time} = \frac{(Force)(\Delta Distance)}{Time} = \frac{Work}{Time}$
A: Energy is a property of a body or system. Things have energy.
Work is a property of an interaction and represents a transfer of energy. Things do not have work, they do work (or have work done upon them).1
This distinction may seem niggling to you---it certainly did to me when I was learning---but it is useful in making communication between humans clear and making the words match the math. It is worth your time to school yourself in keeping the two ideas (energy and transfer of energy) distinct in your head.
If the circumstance are well enough defined you can equate work done with a change in energy (indeed, that is exactly what we're doing in the work-energy theorem), and if you know the time-scale of the process then you can use that to compute the power needed.

1 Thinking ahead a bit we could mention here that heat is another transfer and objects never have heat in physics. What the chemists say is a different story.
A: 
Can I say that work = energy?

No. Because it isn't. Work is the transfer of energy. 

I am trying to calculate the energy/ power required to push a plate up and down in a liquid.

Energy is not the same thing as power. Power is the rate of doing work. 

My approach is to calculate the forces that must be applied by the motor to push/pull the plate based on newton's law (as a function of the velocity, weight of the plate, properties of the fluid, and so on). Then to calculate Work as the product between force and displacement. 

Yes, work is force x distance. No problem with that. 

Can I say that the outcome is the required energy? Or is this a misleading assumption, based on the fact that both work and energy have the same measuring unit?

The outcome is an energy transfer from say a battery to the motor thence to the water. In this case you heat the water and the apparatus.  
