if you want to drive yourself slightly mad you can try going to http://lexin.udir.no/?mode=main-pagesub-mode=searchdict=nbo-maxiui-lang=nbo -> Bokmålsordbok and entering barn as a search term and then clicking on the speaker icon to hear a lady say barnet barn barna in quick succession. For what it is worth, my subjective self hears something like "barn'uh" for barnet and "barn'eh" for barna. In any case, I think in real life there us usually much more context available to help you figure out which is meant.
'-eg-' is commonly pronounced as ei. In these cases, the 'g' is always silent in Eastern Norwegian, and pronouncing it would be incorrect. However, some dialect might still pronounce it, but since this course teaches an Eastern Norwegian pronunciation, you should not pronounce it as ei.
'jeg', 'tegn', 'regn' ('I', 'sign/symbol', 'rain')
There are exceptions of course:
'lege', 'legg' ('doctor', 'calf (foot)'/'identification credentials'/'lay (imperative)')
i don't know . . . after 6 months, i don't hear a difference, nor do i hear it when i type "barna barnet" into google translate, nor do i hear it on any other speaking services. if there isn't context to tell me which is which, it's a 50/50 guess. i'm okay with accepting that. there's a lot more to learn, and if i never hear barnet vs. barna, that's okay. it won't make or break my language study.