How come the definite article does not end the word in "-en" as usual? Are these some rules with the plural definite forms?
It depends on what the plural of the word ends with. If the plural only adds an "-e", "-s" or the word doesn't change in the plural form, then the definite plural form is "ene" (e.g. Huse to Husene, sko to skoene and Hotdogs to either hotdogsene or hotdoggene)
If the plural adds an "-er", then "-erne" is the definite suffix (e.g. Piger to Pigerne, personer to personerne, etc.)
Also, with things like nationalities and jobs, the pattern tends to be, for example, arbejdere to arbejderne or danskere to danskerne
I get that, but I can't hear the different pronunciations for: pige, piger, pigen, pigerne.... they all sound the same!
Can't this also mean that the girls are studying, rather than reading?
Simply because the verb "at læse" means also that: http://www.ordbogen.com/opslag.php?word=l%C3%A6se&dict=auto
Well in this case studying most likely involves reading. Extension of meaning I suppose rather than a distinct one.
It's more like studying (also in British English, you can use reading if it's in higher education) for a degree than the act of sitting down and studying. "Læser" is generally used for higher education (university level)
Edit: Actually, I just looked and it can also be used for the act of studying (for a test or class/doing homework)
For this and the one that translated to "they are drinking milk", is the sentence the same as "The girls read" or "They drink milk"? Asking because in the lesson previous, it said that things like "Hun drikker vand" was just "She drinks water" not "She is drinking water". Does the implication that the action is actively happening not change the sentence?
It's really late, but I believe they have just 1 present form in Danish. e.g. Jeg spiser can mean I eat, but also I am eating.
Piger = Girls
Pigerne = The girls
Danish adds the definite article (the in English) to the end of a word with the exception of when there's an adjective (Pigerne er dejlige (The girls are lovely), De dejlige piger (The lovely girls))
Is it necessary to use 'de' in the sentence like 'de dejlige piger' or simple ' dejlige piger' is enough? (as far as I learnt from duolingo 'de' means they/those
Yes, it's the same difference as it would be between "The lovely girls" and "Lovely girls" in English. Although "De piger" would be "Those girls", with the adjective there it can also mean "The", as well as "Those"
Also, to distinguish piger from pigerne in pronunciation, listen for the neh sound at the end. Hope it was helpful.
What I really can't understand is when the translation has the article and when not...
Why if I say that "Mænd spiser brod" the translation is "Men eat bread" but when I say "Pigerne læser" the translation is "THE girls read"? Which is the rule to understand when we must use the article and when not in the translation?