I do not think it is, Latin word order is SMIDAV. Subject, modifier, indirect object, direct object, adverb, verb. The verb is always last (unless using the to be).
I don't know how helpful absolute statements like that really are. Even in the writings of Caesar, which of the classical writers hue closest to the order you outlined, one sentence in six does not end with the verb, and a quarter of those that do put the verb last show the object before the subject, and presumably many more in some other way deviate from that pattern. In the writings of Cicero, fewer than half of the sentences end with the verb.
Latin word order is actually extremely free, but I guess the one you've described is the most common one.
Essentially yes, that is a grammatically correct sentence. I must say that "frater domi scribit" seems more natural to me, though I don't think your translation should be taken as wrong.
It's all right.
The classical prose writers, who have generally served us as our models for good Latin style, tend to avoid ending a sentence with an adverb or adverbial expression, so you will struggle to find sentences structured that way in the works of Cicero, Caesar, Livy, and so on.
In the pre-classical comedies of Terence and Plautus, however, such examples abound (a quick search turned up around 20 sentences ending with "domi"), and in fact it is on these authors that many of these short, colloquial sentences that we are working with on Duolingo seem to be modelled.
I think many more dialects of English use 'father' and 'mother' as title names than allow 'brother' or 'sister' to be so.
I've got a question! Is it possible to say "Frater in domi scribit", since "Soror in urbe est" is correct? When do we use "in". Thank you.
"Domi" is the equivalent of "in domus". There is not, however, an equivalent to "in urbe".