The proposed sentence isn't the best way to translate it, I just tried to find a good example where this phrase could be used. On the other hand, there's a really good Russian proverb with a meaning close to the English one: "В гостях хорошо, а дома лучше" (~"It's good to be the guest at someone else's house, but it's much better to be at home").
No. As a rule of thumb, Prepositional case can't be used without a preposition (makes sense, doesn't it? :) ). Дома is some abnormal noun form and I can't recall any other noun tranformed this way to mean "at smth". So just remember it without trying to explain. ;) Btw, Prep. case of дом is доме and Locative is дому, but this particular word has its own peculiarities, look'em up here: https://en.wikipedia.org/wiki/Locative_case#Russian
Well, it's a bit easier, but still tricky: for some masculine words and feminine words ending with ь in the singular you have to use a special case called Locative. It is always preceded by в or на (while Prepositional has some some other prepositions like о and при). For masculine words it takes the ending -у and for feminine ones it is -и. Also the ending has to be stressed. You can read more about it in the skills Around you and Instr.Case. :)
I read elsewhere that дома as it is used here is an invariable adverb of location, similar in use to далеко (far away). It's not a usage we see in English, I don't believe. It doesn't have a case or gender or number - it's an adverb.
I believe that's why there's no em-dash — between место and дома. The curious thing about it is that usually adverbs attach to verbs, but since there's no present tense of "to be" there's no verb here - but Дома is still an adverb.
"Я дома" (I'm home) is right, saying "Я в доме" (-е because it's the prepositional case) means I'm "in a house", without specifying which house that is.
"Я в дом" could mean 'I'm going to a/the house" ("Я иду в дом", omitting иду), but it should be implicit in the context which house that is. "Я иду домой" means "I'm going home".
I put "my space at home"; i.e. a specific room within the house, like a den, a bedroom or a mancave, identified as being personal. It was marked wrong. I've read the rest of the comments but (and i could be wrong) there isn't an explanation of why it CAN'T be this translation.