I used "Barnet mitt" because I thought 'barn' was a neuter noun, and was using the indefinite form of my (mitt) instead of mine. Am I still correct in this, or am I just misunderstanding something?
Barna or barnene are interchangeable, it's just two options that apply for many nouns. The -a version is considered a radical form (working class, rural areas) and -ene proper speach and moderate (conservative). It's mostly a dialectical choice, but newspapers and official documents usually chooses the moderate forms. (The newspaper Klassekampen, indeed a radical paper with communist/socialist background, use the radical forms of any word.)
Very interesting! The same, then, applies to jenta vs jentene? Is duolingo teaching us a very informal Norwegian?
Jente is not one of those, you have to say jentene. I need to correct myself a bit on my previous statement, it is the neuter nouns this applies to (like brød, hus, slott, bord, barn bread, house, castle, table, child)
I'd say Duolingo prefers formal, but in Norwegian we don't really have a formal and informal, either you can write it or you can't. There might be certain social stigma connected with writing one way or the other, for the layman. Professionally there is much stronger pressure to appear moderate (as it is considered a neutral choice), unless of course you intend to be radical. The official broadcaster NRK and most of the bigger news agencies all prefer moderate (into conservative) and that understandably sets the standard for others. Spoken language has no restrictions whatsoever, only NRK demands of certain hosts/announcers that they speak after a set standard that is moderate bokmål (or nynorsk, the other official Norwegian language), not allowing dialects or riksmål.
Let's see if I got this correct: The number of the possessive always agrees with the noun, so in the very least, since children is plural, you'd use "mine" instead of "mitt". (and since "barn" is neuter, you'd never use "min"!) In addition, "barna" is the definite plural form, and if the possessive is in front of the noun, you have to use the indefinite form, "barn". All in all, if you follow the rules, you either end up with "mine barn" (indefinite form) or "barna mine" (definite form), which are equivalent in meaning.