The potting mix looks very wet, my suspicion is that that is the cause of the seedlings falling over (if there is a constriction at the base of the stem it is an indication of a fungal disease called "damping off" which is encouraged by overwatering) as well as the fungal growth you are seeing. If there is no constriction at the base of the stems and the seedlings are leaning as opposed to collapsing then it could be because they aren't getting enough light.
I think both things are related more to the growing conditions independently but you could try posting to the NGA fungi forum here to see if the growths can be identified:
https://garden.org/forums/view...
