Learn Dutch: De verzorger weet dat de hond z'n halsband kwijt is. - The caretaker knows that the dog's collar is lost.

This is a colloquial way to show possession, matching the focus of this lesson. It uses the possessor (de hond) followed by the unstressed possessive pronoun (z'n) and then the owned object (halsband). It translates literally to "the dog his collar", which means "the dog's collar". This structure is very common in spoken Dutch but generally avoided in formal writing.

Could de hond be the subject who lost his collar, rather than de hond z'n halsband being the entire subject?

Yes, this sentence is actually wonderfully ambiguous! It can be parsed in two ways. One uses the colloquial possessive we are learning: [de hond z'n halsband] is kwijt (the dog's collar is lost/missing). The other treats de hond as the subject of the expression iets kwijt zijn (to have lost something): [de hond] is [z'n halsband] kwijt (the dog has lost his collar). In a subordinate clause like this, both interpretations result in the exact same word order and essentially the same meaning.

Would we ever use d'r instead of z'n for an animal?

Yes. If the animal is biologically female or the noun is strongly associated with the feminine gender, you would use d'r (the unstressed form of haar). Because hond is grammatically masculine and usually assumed male unless otherwise specified, z'n is standard here. For a cat, which is grammatically feminine (de kat), you might hear de kat d'r halsband.