It is probably just a difference between the true German phrase and the English equivalent. Duolingo tends to translate common phrases to the nearest English meaning--like, "Ich drücke dir die Daumen." This is more literally in German, "I will squeeze my thumbs (for you)," if I remember correctly, but Duolingo just translates it to "I will cross my fingers for you." The same meaning, but a different colloquial phrase. So here, it is probably a similar but smaller difference. Hope this helps in some way!
Yes. Duo's way puts more emphasis on the "alles," though. "Ich soll alles machen" is a very matter-of-fact way to go about it, "oh yes I have to do everything, I should get started," while "alles soll ich machen" puts more emphasis on the fact that wow, I have to do everything. Helfen Sie mir, Gott--alles soll ich machen. ;-)
It's typically the subject or a time expression that goes first, but in cases where you really want to put emphasis on something, yeah, you can put it first in the sentence.
Emphasis in English is a little more arbitrary--something at the end will get a little more attention, but ultimately, it's just the intonation that bears most of the emphasis. You can take a sentence like "I have to do everything" and just draw it out in different ways to imply different things.
I have to do everything = it's only me who has to do this
I have to do everything = there's no other option for me than to do everything
I have to do everything = I have to do it all, not just think about it or plan for it
I have to do everything = there's not a single thing I can leave undone
In short, we just use intonation to place emphasis. It's often the same in German, but putting something first in the sentence is just another way to emphasize.
Is this an idiomatic moan? Like something you'd say when one member of family forgets to do something, the of other does his job poorly and then the speaker has to everything?
Close. Your situation would involve "have to" -- Alles muss ich machen!
Duo's sentence has soll (supposed to) -- so the moaner is complaining that they're the only person who gets assigned chores in the household and everyone else gets to sit around on the couch playing with their phone.
Or they're the only child who has chores and their younger siblings don't have any and "I'm supposed to do everything around here".
Yes Gaviota, to be supposed to do something means you should, or are meant to, or ought, to do that thing. This is a quite different sense from "I suppose something" which means I think that thing may be true. There's no sense of obligation here. The two meanings of "sollen" seem similar to these two meanings of suppose.