I'm really struggling with getting to grips with the rule. I get the possession - no problem.
But still struggling when it comes to plural and possession.
So - should an apostrophe be put in after the s on others here?
From becoming popstars, to styling each others' hair
No, "other" is a singular noun when used this way. Compare:
(1) "They like each other".
(2) *"They like each others".
(3) "They like each other's hair".
As you can see, (2) is ungrammatical. That demonstrates that "each other" is a singular noun phrase, so the possessive -'s suffix must be used as shown in (3).