Cultivating Fungi in SoCal (Southern California)

What fungi can I grow in Southern California?
I am growing a food forest in Hollywood, Los Angeles, California. Our USDA zone is 10b. With our mediterranean climate, it can be very wet in winters and very hot and dry in summers. I love mushrooms, and fungi is one of the last layers of food forest I haven't covered. With the heavy rains, I see a lot of mushrooms popping up, but I don't know what they are. Several branches have fallen from trees, particularly Plane Trees, and I was thinking of using the logs as a fungus growing substrate. Any suggestions of what to grow?

I am also interested in any tips one may have about growing fungi in a Mediterranean climate. How do you retain moisture in your growing substrates? When do you harvest?

I already tried 3 times to grow mushrooms. On wood chips, wood dust, and trunks. All failed without producing a single growth. The mushrooms that we know and want to eat grow when it's warm and wet, but on my property when it's wet it's too cold and when it's warm enough it is completely dry.
I thought that it would be way more sustainable to eat what grows naturally and I have downloaded entire Mykoweb website:

Mykoweb - The fungi of California

However, even with few thousand quality photographs this website has, I'm still finding mushrooms that I can not identity and I'm not going to take a risk for something that I can live without.
So far I have found only one mushroom that I was able to identity and that is considered edible and good: Clitocybe nuda, but was still a little bit afraid to eat it. If it grew by itself it would make sense to propagate it on my property.

Cristobal Cristo wrote:The mushrooms that we know and want to eat grow when it's warm and wet, but on my property when it's wet it's too cold and when it's warm enough it is completely dry.

I've found some success in Colorado using microclimates.  Oyster pegs in fresh aspen logs left to myceliate under shade cloth close enough to the yard to get overshoot from the sprinklers for a year, then mostly buried in the dirt and mulch as contour edges where the sprinklers could reach.  Wild Oyster spores on aspen stumps near rain gutters with no irrigation.  Garden Giant spawn mixed in with manure and wood chips around a tree at the edge of the irrigated lawn.  Spreading wild Shaggy Parasol mushrooms in a small grove of trees and shrubs with no irrigation.

I've had plenty of failures as well.  I have a long way to go, more experimenting and research to figure out the right microclimates for the kinds I want to grow at our new place.