For "people", it is plural "mouths". To see this more distinctly, supposing you wrote "some people covered their mouths". Now some mouths are covered and some are not. So you can't talk about a single mouth. But your second example should technically be singular, because "everybody" is singular. The confusing thing is you have a "their" in there, which looks like it should be plural, but it actually isn't -- it's a form of so-called singular they, standing in for "his or her". Also you have a past tense verb that isn't marked for number, but it is actually singular too. To see this, consider a present tense verb that is marked: you say "everybody is here", not "everybody are here". So both the verb and the possessive adjective are singular, and since we are talking about a single person at a time, they (singular) only have one mouth, thus "Everybody opened their mouth".

Hello Rafaela, People is plural ( you could change it for 'they' ) so the sentence completion should be plural - their mouths or their faces. The same goes for everyone/everybody Best wishes Bob

Compare to two boys: their mother => They are brothers with the same mother. their mothers => They are NOT brothers. Each has his own mother.