When ‘euer’ takes an ending it drops the second ‘e’, in this case it results in ‘
eure Kinder’ (not ‘euere’). Otherwise, yes, ‘deine Kinder’, ‘eure Kinder’ and ‘Ihre Kinder’ should all be accepted (and I think they are).
By the way, this happens (optionally) also with ‘unser’ (so, nominative feminine can be both ‘unsere’ and ‘unsre’ ).
The form with the e is an accepted variation of the standard, see here: http://www.canoo.net/inflection/euer:Pron:Poss:2nd:PL and also https://www.duden.de/rechtschreibung/euer_Anrede_wessen .
I didn't know that. While studying grammar most resources said that the elision was optional with ‘unser’ but mandatory with ‘euer’. I suppose ‘euere’ is just much rarer (as a matter of fact, my browser's spellchecker insists ‘euere’ doesn't exist).
Well, we should report it then.
"eure Kinder" uses the informal address ("ihr", second person plural, "y'all's children").
"Ihre Kinder" (with a capital "I") uses the formal address ("Sie", can be either singular or plural).
The sentences Duolingo is showing me here are "Your children" - "Ihre Kinder". That way I can see that Duolingo originally wants me to read a capital "I" here. If I don't know this - and I can't know it, because "i/Ihre" is capitalised one way or another because it's at the beginning of the sentence - then "ihre Kinder" (not capitalised) can also mean "their children" or "her children".