"I come from Australia."
I wouldn't say that it's more natural, "shusshin" is used quite frequently.
As a translation, it's correct. But if you want to tell that you are from Australia, きました or きています sounds natural.
I would argue that OP's translation is slightly incorrect, if they are intending to translate "I come from Australia." Directionality of 行きます and 来ます in Japanese is really important, beyond just sounding natural or not. The directionality of 来ます is toward the speaker, meaning 「オーストラリアから来ます」can only make sense if the speaker is not in Australia but referring to something else that is, i.e. "It is coming from Australia." 「オーストラリアから来ました」makes sense because, wherever the speaker is, they went from Australia toward where they are now.
は wouldn't mark Australia as the subject of the sentence, it would have marked the topic.
は is not present because no particle is needed between the place of origin and 出身.
Alternatively, albeit a little more "pedestrian", you can also say 「オーストラリア人です。」"I am Australian."
No, it doesn't. Where would they go? The only one that could make sense (to me) is オーストラリアの出身 but as I understand, it's not common practice.
I wrote オーストラリア出身だ and it was marked as a mistake by Duolingo. If it's actually a mistake, can you please explain to me why it is?