haha! I was learning French and German at school in the same years. (which everyone said I was weird, because nobody likes learning languages in my country..it's like the new maths!) and my French teacher asked 'what's nose?' and I very nearly said 'die naese'! XD (Didn't help that she was also my german teacher in the first year of senior school!)
There is no rule for when to use "en" or when to use "et". You will have to learn them as you learn the nouns. It is similar to French where there is also no rule about when it is masculine or feminine. It's even hard for native Danish speaker to remember which one to use, so don't worry!
That is wrong. "Manden" means the man, "en mand" means a man. Both "mand" and "kvinde" are n-words, so both use "en" not "et". In any case "en manden" would also be wrong, it would mean: the a man.
Instead of adding a word like the we simply move "en" or "et" to the end of the noun to make the definite form.