re: "How can one prevent baby/recently sprouted plant disease?"
With knowledge & proper care.
re: "When one desires to germinate seeds in the seedling tray, he/she has to keep a humid and warm environment."
Depending on the type/s of seeds that you are trying to germinate, you may not need a warm & humid environment. Most garden seeds germinate fine at 68°F when planted in open soil (i.e. no cover required on top of the seed tray). Just plant, water, & mist occasionally. I never use seed tray covers & I never have problems with seed germination or disease caused by fungus or other undesirable factors that favor humid conditions.
re: "Other than sterilizing the soil, how can one prevent fungus growth on baby plants?"
Start with good soil & then allow air to circulate around your seedlings (don't use a seed tray cover--it isn't necessary in most cases).
re: "Also if you sterilize the soil will you have the same germination rate?"
Seeds germinate fine in sterile soil, however, beneficial microbes often help plants grow better due their symbiotic relationship to plant nutrient availability.