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I am sorry the mulch is not working out as you hoped. Normally mushrooms occur where the ground is staying too wet. If you are using a sprinkler system, you may want to consider watering in long once a week spurts (which will let the top of the mulch dry out some) rather than short daily spurts. You can also treat the areas with mushrooms with fungicide. This will kill the mushrooms. If it helps, mushrooms are a sign that the soil is very healthy.

Ground up leaves make a superb mulch. You'll want to mulch twice a year, less (a couple inches) in spring, and 4 - 6 inches in fall. Here's some general mulch information: https://www.gardeningknowhow.com/garden-how-to/info/whats-the-best-mulch-for-my-garden.htm
and a couple of articles on leaf mulch:
http://www.extension.umn.edu/garden/yard-garden/soils/options-for-disposing-of-leaves/

I would suggest using a search engine and use "buy black glass mulch in San Francisco Bay area". This should help you find sources.

I can't find anything in the literature that suggest tea tree (melaleuca) mulch is poisonous to dogs and cats. Tea tree oil in high concentration can be dangerous; this may be how the tree mulch became associated with toxicity. The only mulch that is consistently listed as toxic is cocoa-shell mulch. Of course, any kind of wood mulch that is ingested frequently by a dog can make them sick; if you have a dog that just won't leave the mulch alone, you might want to consider preventing them from reaching the mulched area, or using large river stone mulch (even small gravel can cause damage to a dog if it's swallowed.) I did see one dog-breeder forum that recommended using tea tree mulch. This article has some more general information on mulches that might be useful to you: https://www.gardeningknowhow.com/garden-how-to/info/whats-the-best-mulch-for-my-garden.htm

They are likely either midges or fungus gnats. You can treat the mulch with a fungicide and get rid of them quickly or you can just lay off watering for a little while to allow the mulch to dry out some and this will take care of them as well.

Below is an article on pet safe mulch that will be helpful to you. It may even cause you to reconsider mulching your dog area given that dogs can choke on mulch or have an allergic reaction to it. Mulch may also not be very gentle on your dog's paws. If you do decide to pursue the mulching route, here is a cool mulch calculator that you can use to figure out just how much you need: http://www.landscapecalculator.com/calculators/mulch

I have read that perennial ryegrass is ideal for dogs to run on because it grows low, fills in fast, and recovers quickly.

Here is the article on pet safe mulch:

https://www.gardeningknowhow.com/garden-how-to/mulch/keeping-mulch-safe-for-pets.htm