In every case, the referent to the pronoun is plural, so use the plural.
1. Nobody knows what is inside people's heads/head
2. We have green hats on our head/heads
3. They shouldn't go there. There it is so cold that their nose/noses will come off. (everyone has one nose)
4. Just for a laugh. They will be able to stick a sticker on their head/heads
5. The hostages were tied so that they couldn't move their hands. (we have two hands :))
6. The hostages were tied so that they couldn't move their body/bodies.
Student or Learner