What is it and what should I put on them?
The first photo looks like a leaf spot fungus. Here is more:
The next two photos look more like bacterial leaf spot (halos around the spot):
Your article states there is no remedy for white mold, so my garden is doomed, I want to cry! After clipping back my plant what can I do for the soil to kill or discourage mold growth?
Does your whole garden have it or just that spot where you cut back a plant? You can always solarize the soil to kill pathogens. That entails covering the area with clear plastic and weighing the edges down with bricks. It takes several weeks with sunny days. Best done in summertime. If it's not too large an area, you can remove and replace the soil. Where you cut back a plant you can treat it with a fungicide to see if that helps. However, the article says it's best to remove infected plants.
My dog rolled in this, it was covered in flies and other flying bugs. It smells so bad I almost vomited.
It could be. Here is more information about it and other things in the lawn with an odor.
It is initially puffy and almost sticky, sort of looks like vomit. I’m guessing that it is a fungus of sorts but it only appears on my mulch. Is this harmful to animals and plants?
It is a common mold that mulch tends to develop now and then, which indicates it is decomposing. You can ignore it if you want. Or spray some vinegar.