The falling over thing is called "damping off". Microwave your cupful of damp soil and get it steaming hot. The heat will kill the damping off spores. Let it cool completely before planting your sprouted seed. Fans also help by keeping the moist air from building up around your plants. Damping off fungus just loves stagnant, humid air!
I've only seen that in the cloner...then again, I'm a hydro guy, so can't say as I've played with soil much...