"He spends a lot of time with his children."
Translation:Han tillbringar mycket tid med barnen.
Shouldn't "Han tillbringar mycket tid med sina barn" be the only correct answer? "Han tillbringar mycket tid med barnen" is not specifying that it's his children, which I would say makes the sentence an incorrect translation.
Swedish very often uses the definite form when English will use a possessive pronoun. If you say in English "I'm brushing my teeth", the best and most natural sounding Swedish translation is "Jag borstar tänderna". And the other way around, if you'd translate "Jag borstar tänderna" into English, the best translation would be "I'm brushing (or brush) my teeth", not the teeth. "Jag borstar mina tänder" is not wrong in Swedish, it's just not the idiomatic way of saying it. – In this particular case, both ways sound pretty natural, however I still think "Han tillbringar mycket tid med barnen" is a little better. You're right that the sentence does become a little ambiguous, but the first interpretation of it would still be that it's about his own kids.
this is rather a rule of english. you cannot have body parts or family members without possessive pronouns in those cases. but you can in swedish.
You can use that without possessive pronouns in English too. Body parts is some other topic.
Well, body parts can be said without possessive pronouns in English, but it sounds very clinical and impersonal. Outside of a medical context it's not really encountered.
No, that would imply his children live someplace else on their own, which is not implied.