"Ei aduc copiilor hamsteir" is considered correct. Shouldn't the form "le aduc" be necessary? I know that the meaning is understood without the "le", but I got the impression that you can't omit the unstressed pronouns.
Here it is correct with or without "le". But don't make it a rule. It is hard to learn Romanian from rules. They are too many and have too many exceptions.
I actually find quite natural to omit the dative pronoun even when the indirect object is a person.
Would "Ei le aduc hamsteri copiilor" be correct as well? Does it matter whether "copiilor" comes before or after "hamsteri"?
No, it does not matter. Report your answer if it not accepted.