Back in a previous life, I fixed cars for a living. If I had a dollar for every car I saw with a whining timing belt, I would be retired...
I was almost always caused by the belt being too tight. The teeth on the belt trap air as they engage the sprockets, and the sound comes from the air escaping as the belt compresses. Many manufacturers use formed steel sprockets and drilled small holes in the bottom of each leaf to allow air to escape.
Any way, the whole "x amount of deflection under some type of finger pressure" is a horrible standard for setting tension. Try this is instead, you should be able to twist the belt 90 degrees with minimal to moderate pressure using just your thumb and 1 finger on it's longest span. You will quickly know if the belt is too tight because it will require extra effort to get it to 90 degrees.
It is normal for a belt to flap or vibrate when turning off the machine because the spindle is trying to ever run the motor. Timing belts vibrate more than v-belts during shut down because they are less rigid. It is also normal for a belt to flap or vibrate at high speed, and once again a timing belt is more prone to this than a v-belt, and a timing belt will set up a more severe resonance than a v-belt. Well designed machines use idler pulleys against the timing belt to stop this vibration, if you study a variety of machines, you will see that v-belts will often run several feet between sheaves, but timing belts almost always have multiple idlers. My big planer for example has over three feet of span between the cutter head and motor sheaves and there is almost not vibration while running.
Turning a timing belt (or any belt really) around a very tight sprocket will also cause unwanted vibration because the belt is trying to spring back straight as it exits the sprocket, this is magnified if the belt it too tight. This can be reduced by loosening the belt enough to reduce the high tension, but not so much that the belt can gain large lateral momentum.
If the sprocket you purchased is smaller than the factory one, it would not surprise me at all if it vibrates and sets up harmonics. I would start with getting a good belt (I would do this even if there was no harmonics, good belts make a big difference in finish). Stiff machine belts that are designed to not stretch will have a low tolerance for small sheaves. Check the manufacturers web site, they will publish minimum sprocket diameter.
Lastly, you might want to double check that the pulleys are actually properly made. If the teeth are not the correct size, it will always cause noise and vibration. Just because some random guy on Ebay can program a CNC to cut a sprocket, does not mean that the same guy knows the first thing about belts and pulleys - there is a good chance he just copied measurements from some other pulley (or even worse, took measurements from the belt) rather than starting with pitch going from there. The choice of a trapezoidal tooth is also a really poor one, round tooth belts are quieter and smoother - you do not need the improved power transfer of a trapezoidal tooth. Just looking at the picture of the sprocket, the leaves look too thick to my eye. Normally the gullet is the same or wider than the leaf, your pulley looks to have a much wider leaf than. Compare to a commercial sprocket
http://www.designatronics.com/images/center-distance.jpg
