Potting soil / compost often breeds mushrooms. I have some under one of my trees too. They show up in damp weather. Mine look different than yours.
I have no idea what the eggs are. They are under the leaves too? Figs don't have many pests. Mine get spider mites in summer, but I've never seen those blue eggs. Try the Name that plant forum. They might know. or do a web search.

They are called artillery fungus, they shoot a spore packet that looks like an egg that sticks to things.