Anyone know if kvinne and queen are etymologically related? They sound similar and 'queen' as far as I'm aware is not related to any of it's latin language counterparts.
Yes they are. Someone asked this on Swedish (woman is kvinna) and since the North Germanic languages are essentially a dialect continuum, the words kvinna and kvinne are almost certainly related; therefore both are related to queen (originally queen was spelled cwen before pronunciation changes and adoption of Q)
En is masculine and ei is feminine. You can replace feminine with masculine though. Ei kvinne and En kvinne are both correct.
Du er ei kvinne and Du er en kvinne are equally correct. However, in a listening exercise you have to type what you heard and not what you usually associate with that noun.