Yes it's OK to use outdoor soil, and the mushroom is, as Richard already noted, a sign that yes, you do have fungi in there.
I don't see the moss in your picture, but you may be waterlogging there.
Soil in pots is different from soil in the wild: there, moisture can diffuse into the environs, in the pot it's stuck. Add to that that your setup doesn't promote evaporation by heat. So you need to make sure you have sufficient aeration in there, especially if the clay content is high, or you will end up with anaerobic conditions in the pot, which will disturb nutrient cycling, at the least.
The way the soil looks on your pic (pretty heavy), I'd be tempted to repot, mixing the rootless portion of the soil with inorganic and organic additives to get it more "fluffy". Personally, I like pumice and coco fiber, just for an example. Both can be a mix of different things too, i.e., add biochar (max. 10-15% of the total volume) and smashed pottery shards to the pumice, or any plant material that is more on the woody side - stalks of plants that died down standing will decompose faster, chipped branches of bushes or trees (NOT conifers! They're too far along the successional line for cannabis, also contain terpenes that prevent fungal colonization, so would have to be aged to the point of not smelling of "pine" anymore to be beneficial) will take longer.
HTH!