Does changing gear relation produce different speed at same applied power? When you are starting the ride, I potentially believe ("know?") that with a smaller plate(cog with fewer teeth in the rear cassette), you obtain more velocity than with a bigger plate(cog with more teeth, in the rear cassette), applying same human power consumption on both cases. In first case (the correct way of starting the ride), the chosen plate radius is small, so it weights less and offers less resistance.

Once bike is running at a normal speed on flat terrain and gear's start resistance has been overcome, I want to know if I will get more speed with a smaller tooth rear cog than with a bigger cog, applying same human power consumption on both cases.  Changing relations would of course mean I should ride faster of slower in order to waste my same power in all cases, basically because the different resistances of the plate, which depends on its chosen radius mostly.

So let's say I want to be using 200 Watts. Would different gear relation setups affect the output speed?

If my statements above are wrong in first please, correct me.
Yay! Another physics question.

Restating the question: if the rider power input is constant, does the bicycle velocity change if the drive-train gear ratio is changed?

Let's assume the road surface, wind speed, gravity etc. all remain constant. Bike and rider are on a surface with no gradient.

Short answer is that constant power results in constant velocity regardless of gear ratio. Gear ratios change angular velocity and torque but because the product of the two is constant power remains the same.

Power at the rear wheel = power provided by rider - power loss in drive-train.

If the power loss in the drive-train does not change with gear ratio selected, the power at the rear wheel does not change and hence velocity does not change.

Of course, the power loss in the drive-train will depend on the gear ratio selected to some degree. I believe the chain travelling more slowly around a larger sprocket consumes less power (I'm not sure on this so somebody please correct me if I'm wrong). 

However the power needed to overcome air resistance and wheel rolling  resistance is much greater than drive-train losses (if the velocity is not very low), so a change in drive-train losses will not effect velocity very much.



Update: addressing this statement from the OP (edited), as I think it's crux of the misunderstanding.

...with a smaller [rear sprocket] you obtain [higher] velocity than with a bigger [rear sprocket], applying same human power...

That would be true if human power was replaced with pedaling rotational speed.