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 theof 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, however, beneficial microbes often help plants grow better due their symbiotic relationship to plant nutrient availability.