"A hungry animal is eating my child's clothes."
Translation:Et sultent dyr spiser klærne til barnet mitt.
'Mitt barns klær' will do. It is also possible to say 'mitt barn sine klær' with the same meaning ('sine' changes with the noun, i.e. 'mitt barn sitt rom', 'mitt barn sin bok'), however the entire statement requires a confirmation from native speakers. I have never met the reflexive possessive expressions "in the wild" except short mentions in reference books on norwegian grammar.
The second structure you mention is called garpegenitiv. It's generally not considered good form in writing, perhaps because it doesn't exist in Danish, which used to be the language of the upper classes, but it's still acceptable, and sometimes advisable. When the owned noun already ends in an -s (or s sound) before being turned into a genitive, using garpegenitiv can add clarity and ease of pronunciation:
Anders sin bok
It also sees some use when what's owned is a longer unit, for the same reasons:
de eldre norskelevene på Duolingo sine lærevaner
Outside of those two contexts, it's rare in written language.
An interesting article in Nynorsk:
As usual you have found the right words to make things simple and clear. Thank you!
As for the article I made a couple of honest attempts to read it but haven't made it to the end. It takes a good deal of intuition to see and recognize that
mye let alone the need to reread each sentence twice or thrice in order to make sense of it for myself )
it's a whole separate letter in the Norwegian alphabet, not a combination of a and e. The best solution is to go to keyboard settings on your phone and add Norwegian as one of your keyboard languages. Duolingo will auto switch the language appropriately depending on the exercise you're doing (though you also still have manual control of switching the active keyboard language when you wish). The same ability to add and quickly/conveniently switch keyboard languages is available on computers.