Concerns about broken washer found in water supply
First guess, it's just junk from the pipes: you didn't properly bleed the system after having used a shut-off; remove aerators, run water.

Next, a faucet with a sprayer has a flexible line; each end having a grommet that may have deteriorated (as with all other flexible lines).

Lastly, if the valves are old enough, their packing may have deteriorated to the point of breaking (but that seams unlikely if they still actually shut-off the water). Most likely it's just little bits of the valve's seat that were near pressure-cut through, and finally came lose when you actuated it.

That a valve leaks when 'out-of-normal-position' does not surprise me in the least. Welcome to the world of plumbing: now ya did it. Some valves have a nut or a screw that you use to snug-up the seal. With most modern valves you slightly tighten the entire assembly with two wrenches.

You say the particles are black; that precludes my last guess that it's pieces of your HW tank's dip tube (these should be some shade of white).

Solution:

Remove all aerators in the home. Clean them while you bleed the system for a few minutes from each of those fixtures. If the problem reoccurs, something is deteriorating and ending up in your water. This should be found if you want to eliminate any (IMO, completely negligible) health concerns. If it doesn't, now you know the steps to take when you use a shut-off.