Yep, just fine to compost. If they weren't, everything would eventually become toxic, which of course it hasn't. Somehow Nature has ways of sorting things out. It's almost like magic! It sounded to me like the only reason you were asking 'poisonous or not' was in terms of composting rather than eating. But I second ericwi's advice if you are looking for edible ones. There are some excellent resources online. My state conservation dept. actually put a fact sheet with several dozen edible types that grow here. I don't rely just on that but online resources like mushroomexpert.com will give you all sorts of tests so that you know 5 ways to Sunday that a shroom is safe to eat.

Mushrooms are the fruiting bodies of enormous networks of fungi under the soil. They're actually a good thing, indicating that the soil is pretty healthy. You want good fungi down there as they do plenty of jobs for you, like aerating and processing organic material into plant food. You can pick them if you like, the only thing you're doing is removing the fruit from the tree, so to speak. For most mushrooms, I tend to simply ignore them.

Actually magnolias are a classic host for Armillaria infestations, if a body is ever going to see a Hellish example of a honey fungus attack to a planted tree it will be on a magnolia. For example the perfectly fine big-leaf magnolia I grew here for years, only to have Armillaria get up onto the stem one year and pinch the top of the tree off in short order. In other words the fungus kills trees and shrubs at ground level, causing the top to topple. And apparently become more accessible to it, because in the case of a tree with a substantial trunk at least it is now a fallen log, lying onto or much closer to the ground than it was before. Armillaria is susceptible to dryness, needs conditions to be moist for it to do well. Hence producing rhizomorphs for coming up out of the soil and onto stems, like subterranean termites with their mud tubes built to provide bridges between the soil and wooden parts of structures. And the fungus proliferating after it gets under the bark of targeted plants, where it is protected from sun and air.

Twice a week for 10 minutes is not enough for a cool season grass in the extreme heat. Having said that, it seems to work on your front lawn, so why is that? Are the sprinklers identical front and back? Same heads? Same number of heads per zone? Same apparent coverage? You might put out some tuna or cat food cans and collect water for the time your sprinklers run and see if you see any differences. Generally we talk about deep and infrequent watering, but we seldom talk about proper plant selections adapted to different areas. You live in an area of extremely extreme weather with winter time temps in the teens being not that unusual and summer temps above 100 not that unusual. I would tend to lean toward non traditional lawn grasses like wheatgrass, blue grama. These are considered to be prairie grasses, but they do very well in your conditions especially when they are seeded to be dense. They need much less water to survive. If you let them go long enough they will develop seed heads, so some amount of mowing is required. But again, it is less frequent than a typical lawn type grass needs. I think your yard is alive, and will improve after the summer heat breaks. It will be better next year after another year of root growth. Still for long term life it will need much more care than a prairie grass type lawn.