Hello, Alex, and welcome to the Grammar Exchange.
Sentences 1) and 3) are certainly correct as Okaasan states. Since everyone has just one just one mouth, it is clear that in this case, 'everyone' is perceived as one individual.
'Mouths,' of course, would not work since the image would be quite monstrous: that every one person has more than one mouth*. However, 'lives' could also work because the idea could be that, since everyone has an individual life, together they are lives.
There is a problem, as you know, with 'everyone.' While plural in meaning, it is grammatically singular in that it takes a singular verb. But, there is no proper pronoun to refer to 'everyone.' We could use 'he or she,' and 'his or her,' but because it those constructions are so awkward, we find 'they' and 'their' workable and acceptable.
If we said 'his or her life/ mouth' it would be natural. However, with these nouns following 'their,' it seems comfortable to put plural nouns here. Sometimes we can use either.
The topic here is that of 'distributive plural.' We have had similar discussions here on the Grammar Exchange previously. Please take a look at this one:
http://thegrammarexchange.info...24&a=tpc&f=340600179_______
*While the singular 'mouth' goes with the singular 'everyone,' if we use the plural 'people,' we might say 'mouths.' We can say:
People have mouths to eat with.
or
People have a mouth to eat with.