Mushrooms: Mushrooms are the fruiting body of soil fungi. They're harmless (unless eaten, in some cases, of course). They actually mean the fungi are healthy enough to send mushrooms, so this is a good thing. You want a healthy and diverse soil fungal population.

Moths: If they were low-altitude (not more than a few inches above the turf) and small, they were probably sod webworm adults. They're not an issue in reasonable numbers, The sod webworm can damage the grass, but grasses happily overgrow the damage except in extreme cases. No control is required unless you see widespread larvae and lots of damage to the grass blades.

thank you