Is the pronoun 大家 (everybody) a subject only?
While 大家 means 'everyone', it refers to 'everyone' in a group of people that is implied in the sentence, or everyone in close proximity

佢哋唔鍾意大家


大家 in this case implies everyone within the group of '佢哋', which is essentially 'each other'. Also this sentence sounds perfectly natural, I hear this a lot when people are referring to couples who don't actually like each other

大家好


大家 here refers to everyone that you are speaking to, basically 'hey everyone'

大家都唔鍾意佢哋


There is an implicit subgroup '大家' where everyone in that group or everyone here dislikes 佢哋, 'everyone (here) dislikes them'

大家咁話啦


'You too', 'same'

大家 can definitely be used for 'each other', however it is not the only usage, and there is not a direct one to one translation for 大家 in english.

In other special cases, for 'each other' / 'one another' you could use:

'彼此'相愛


to love each other

'互相'鼓勵


to encourage each other

'自相'殘殺


to hurt and fight each other