It's not the genitive here, it's the locative.
Domi is an exception, it takes the locative case, which is a very rare case in Latin.
Domi, ruri, and humi, always take the locative, as an exception.
Domi: home (used as an adverb), at home, in the mother-country (in the home?)
Ruri: in a rural area. (from "rus")
Humi: on the ground, on the floor, on the soil (from "humus").
And: cities, towns, and small islands
also take the locative.
Your theory (or hypothesis, technically,) is correct: 'domi' is always the locative singular, and is used to show where something is.
The genitive singular of 'domus' is 'domus' (with a long 'u'), not 'domi', because 'domus' belongs to the fourth declension. (Note that most nouns in this declension are masculine or neuter but 'domus' is feminine.)
The ablative singular, 'domo' or 'domu', is used to show where something is coming from.
bas sake buxbeze sabeDev
You say 'domi' is always the locative singular.
Not quite completely so. "Domūs" is the form generally taught, and the form that should be used in Duolingo. However, outside Duolingo, you might encounter "domī" as a genitive singular for "domus".