No, they are not interchangeable at all.
Домой means "home, homeward" (as in
"go home"), and
дома means "at home" (as in "stay at home").
Thus, "моё место дома" is indeed correct, because it answers the question "where is my place"? At home.
Now you can just memorize these forms the way they are but at some stage of learning Russian it's just inevitable to need to learn more about Russian cases.
See there are no cases in modern English at all (only some traces like "who"/"whom"/"whose") but you just know what's the difference between "at home", "by home", "to home" etc. - it's just something you've actually memorized - some of information that is passed by prepositions in English is passed by the cases. It's just the way it is.
Well, it's even more tricky, technically, "домой" and "дома" are adverbs, not nouns, so it's not correct to talk of cases but actually those are historically just "frozen" forms of noun
And, to introduce even more confusion, those are "frozen" form of so called smaller cases - one that are not used for all nouns but there are some remnants of ancient, richer than modern, conjugation system.
So, just accept it the way it is ;)