Clustered eggs like that could be some kind of slug. I'd crush them and leave it there for extra protein.

That's an interesting looking growth. Slug eggs are pretty much spherical as far as I know, and these look more cup-shaped. That and the fact that they're different sizes makes me think they might be some kind of fungus rather than critter eggs. Maybe this one? http://www.naturespot.org.uk/species/lemon-disco Though the info says they grow on wood - any wood chips or pieces in your bin?

In general you shouldn't have the bin contents be too wet or soggy. In those pictures it does look pretty wet. You could stir in more dry bedding or fluff up the contents more often. Is there good drainage at the bottom for excess moisture to drain off?

If you have a lot of fruit flies, you can cover up the food with lots of extra dry bedding on top, and/or fasten some mosquito netting or other very fine mesh fabric around your bin (some people use curtain sheers). You can also stop adding new food for a while, a couple weeks is often enough - fruit flies only like their food in the just-turning-mushy stage and it stops being edible to them once it's more broken down.

How strange! It does look a lot like those photos. There are no bits of wood in my worm bin, only coconut coir, shredded office paper, and fruit scraps. It does have good drainage, it drips into a bottom compartment that has a tap. I had recently poured a cup of water over it to leech some liquid out of the bottom. I will definitely pause on the feedings and let it air out for a while.