Great post and valuable information!
I’ve made several dyeing-experiments myself, and would like to add some of my experiences.
(Some pictures of my experiments can be found here: Dyeing pre-colored (grey) PBT caps?)
So far I only used iDye Poly, since it’s the only dye suitable for synthetics that I can get here.
I actually boil the caps, and I never experienced that terrible unevenness as seen in your pictures.
I’m really curious why…
I can confirm that. Depending on the colour, I dilute quite a lot. Otherwise the caps get too dark to fast, unable to control the shade.
I tried it with a few drops in the beginning, but I found that the results without soap are as even/good as with soap. And since I boil the caps when dyeing, doing so without soap spares my the disgusting colourful foam 
I can confirm that. I made myself a special glass strainer that I can lift in and out of the dyeing solution, and by that “stirring” the caps just with the flow/eddies of the liquid without the need of a tool like a spoon.
In the beginning I used destilled water, but after a few runs I decided to try with our tap water.
That’s something that varies a lot from place to place.
Our tap water is very soft, without chlorine, and as far as I know very low minerality in general.