Hello Jamee,
I'm not really concerned with the black roots unless the new roots are doing the same thing. The soil/dirt the plants were grown in color and stain the roots, and they also turn dark through dormancy. Some of those roots may die as well. So as long as the new roots are coming out normal I wouldn't really be concerned with the color of the roots much.
The slime thing on the other hand may be a problem, and could be because of pathogens in your system. I have seen a lot of slime buildup in systems, but never as big a blob as attached to those roots. Considering your reply's that your water has no smell, is clear/translucent, and no foam, I'm not entirely sure you have a pathogen in the system either. For your roots to have such a large blob like that attached to the roots, there should be some sure signs of pathogens in the water. However your not experiencing clowdy water, moldy/musty smelling water, or even foam on top of your water. Considering you don't have any sure signs of pathogens in the water and your plants are looking healthy otherwise, I'm going to say it may be a type of algae. Now you may say if it were an algae it would be green, not necessarily. Algae comes in many colors. Some algae even give off a red, green, or blue glow. So it just depends on the type of algae.
Here's what I would do. Take the plants out one by one and wash the slimy blobs off the roots thoroughly. Set the plants in some water temporarily (half way up the baskets). Possibly even H2O2 water (at the same dilution rate you used), and even let them soak in it for a few hours. While the plants are all out of the system, give the inside of the tubing a quick scrubbing and rinse them out. Put it all back together, and wrap the tubing in a light proof material to block any light from getting in the root zone. If it is an algae, all you need to do is block light. I would also change the
nutrient solution to a fresh batch, clean out and rinse the reservoir, and if the reservoir isn't light proof, I would light proof that as well.
Now, you have 2 possible sources of pathogens. One from the plants roots themselves because they were grown in dirt. Soil born pathogens attach themselves to the roots, and you cant wash them off no mater how well you wash off all the dirt/soil. The other is from your water source itself. Since your using ground water and not sanitizing it before use, it can contain soil born pathogens.
Even if your not seeing pathogen issues right now, and the gray blob is just algae, you should still take precautions ageist pathogens in the future. Especially since your water source can contain soil born pathogens. The easiest and safest way to do that would be using an aquarium UV light sterilizer. UV light treatment systems come in all kinds of shapes and sizes, and are used for everything from sanitizing household well water, to aquariums and ponds. Some are submersible, and some are external.
Just make sure you get one that is rated for the total amount of water volume you have or more. Here are a few links to some so you know what I'm talking about. The UV
lights will also help keep algae down if your system isn't 100% light proof.
Green Killing Machine Internal UV Sterilizer with Power Head
Aqua Medic Helix Max UV Water Sterilizer, 5 Watt
aquarium uv light sterilizer (Amazon)
Now with all that said, you could be experiencing
biofilm buildup as well. Which is like algae in the scene it's a slimy film, however it can thrive even withough't light. Using the UV light filter will kill the microorganisms in the water that collect on surfaces to form the slimy biofilm substance.