'gerne' is kind of to represent she really wants it, or it makes the statement more polite. Both 'pigen vil have en kat' and 'pigen vil gerne have en kat' are correct, however the later just sounds nicer and it makes you seem more nice and polite
I think so. why? ^^ Because I've "hardly" learnt that before with German language, they also have the same word and similar grammar uses! (Notice how they both are related in origin, geography and history)
Why is the "e" or the "er" silent most of the time at the end of a word in speaking Dansk but very strong and clear as with "have" in this sentence? Is there any rule or is it just word by word learning?