Why do you sometimes put van but other times just ons?
'Ons' is the object pronoun ('us' in English), 'van ons' is the possessive pronoun ('ours' in English). See also here.
so is like 'our book' vs 'the book is ours' ?
Yes. Pretty much...!
So "Van ons" translates literally to 'from us'? If so, "De hond is van ons" translates literally to 'The dog is from us', can be understand as 'the dog is a part of us' and therfore it is ours?
The dog belongs to us. Why is my translation wrong?
"van ons" literally translates" of us, or "ours" whereas "ons" is "us" so van ons" would be "the dog is ours" whereas "de hond is ons" would translate "the dog is us" or "we are the dog" which I'm sure is not what you want to say :P
“De hond is ons” is wrong Dutch, only “De hond zijn wij” is possible (though it still sounds odd as dog is singular)
How would you say "the dog is from us"?
As in: a gift? Q: Who gave us the dog? A: The dog is from us.
I don't know, but interesting question.
What about "this is our dog"? Equivalent to "this dog is ours"?
That'd be "Dit is onze hond"
van ons vs van onze, anybody? I don't get that.
Onze banaan = our banana
De banaan van ons = the banana of ours
Ons huis = our house
Onze for de woorden and plurals, ons for het woorden and van ons for both.
it didn't accept when I typed "it's our dog" which is also what an english speaker would say. was it not accepted because it was not as literal a translation?
That English sentence has a Dutch counterpart, namely “Het is onze hond”, they use different words and have a much different sentence structure, the subject isn’t even the same (it vs the dog).
So it is not accepted.
Why is "That is our dog" not accepted?
Because the sentence does not contain “dat”.