Domo: could be dative or ablative. I think in this case, it's ablative, as the ablative is the case expressing a move from a place.
Can we say "a domo" or simply "domo"? (with "a" meaning away from)
Domi = at home. Locative, would be literally "in domus". So, yes, I understand why it's "at home".
Domum = accusative or genitive, this one I don't understand why it's "from home".
Any noun that supports the locative case drops ad/ab prepositions for motion toward and motion away. So no, you wouldn't write "a domo" for "from home", nor would you write "ad domum" for toward home. In both cases, you drop the preposition because domus has the locative case.
If you are following the other lessons, the same holds true for Bostonia, Philadelphia, Eboracum Novum, and other cities.