Just like with Нет. Both are negative, and implie that "There is none OF ..." The "of" makes it genitive.
"without a mom" would be better... it's strange just having "mom" at the end there
That phrasing is what American English uses to say that a child has no mother - either dead, in jail, or separated from the household for various reasons.
"without his/her mother" would be normal English. I'm curious as to whether the Russian implies "his/her" in this sentence.
In English, if you said, "A child goes to school without his mom", it's understood that "his" is a generic possessive pronoun which can also mean "her" - a reflection only of the default masculine gender found in Romance languages.
Is the same true in Russian, so that Ребёнок идёт в школу без его мамы would be understood to mean "his mom" or "her mom", when you're speaking generically about a child - Ребёнок after all can mean a boy child or a girl child.