when do we use et = a and when en = a ?
That depends on the gender of the noun. 'et' is for neuter nouns, 'en' is for masculine nouns and 'ei' is for feminine nouns.
and how do we know if it is neuter, M or F?
You learn it by heart. Always remember the gender of a noun when you learn it. Always.
Why is it Et barn and not Barnet?
This matter (indefinite vs. definite noun reference) is dealt with in previous lessons of this course. If you refer back to them you will find plenty of exercises that will help you refresh your memory:
Et barn = A child vs. Barnet = The child
I feel the robolady pronounces gråter like grotter with a stress on t