No, "min hund" is "my dog" and "hunden min" is "the dog of mine", but for nouns you use a preposition and don't add the s when the possessor comes second.
"barnets hund" is "the child's dog" and "the dog of the child" is "hunden til barnet".
The preference is to put the possessor first with the s. The dog could be indefinite or definite after the possessor, but if the possessor comes last, then the possessed item must be definite.
Whoa... the "tips and notes" for this unit only discusses possessive adjectives (my, your, our, etc.). There is no mention at all of adding an "s" to a noun; are we just supposed to figure this out from example? Admittedly it's not hard to figure out what's going on, but still, not discussing it first in the "tips and notes" seems like a serious omission. I'm not sure where to put suggestions or error reports for the "tips and notes", so I've put it here.
Could some maybe explain why det is used here and not den? I got confused when seeing the sentence and the sentence "det er kattenes mat".
If det refers to the child, shouldn't it be "den kattenes mat"? Since it is en katt
If det refers to mat, then I would expect "den er barnets hund".
What am I missing?
The word for child is neuter it takes "et" ending and "det" is the right one to use for it. Be careful not to confuse the gender of the person or animal represented by a word with the actual gender of the word. It is "det barn" but "den hund", but this is a set expression "det er" where we don't know what is coming next yet. This doesn't mean "it is" unless it is coming after a sentence with the noun already stated, instead it means "There is" or "That is"
I am sorry. I have confused the issue. Here "det" is not referring to either following word. "det" as a pronoun could replace "barnet" in a sentence after the neuter word has already been used and "den" as a pronoun could replace "hunden" in a sentence after it has already been used. Here "det er" is introducing the sentence as a set phrase, as "there is" or "that is". https://www.ntnu.edu/learnnow/3/grammar/
I wouldn't speculate why it is the way it is, but you are right that there is no noun being referred to by the pronoun "det" in that case.
Also, "det" and "den" can also be used like adjectives with nouns to mean "that", "this" or "the".
If you scroll down here, you will see many examples of the use of the word "den":
http://www.norskengelskordbok.com/en/dictionary-norwegian-english/den and now scroll down here for many examples of the use of the word "det": http://www.norskengelskordbok.com/en/dictionary-norwegian-english/det