A pronoun must agree with its antecedent in gender, person, and number. The gender and person usually take care of themselves, but the number of pronouns is a serious obstacle to correct speech.
“One tells the quality of their minds when they try to talk well”—George Eliot, in Middlemarch. The pronouns their and they should be singular.
“Everybody has something to say which they think is worthy of being heard.” Everybody refers to persons singly, and not collectively. They think should be he thinks, he being the proper pronoun to employ when the gender is not indicated.
“Every nation has laws and customs of their own.” The use of the word every necessitates a pronoun in the singular, hence their should be its.
“Every one is accountable for their own acts.” Use his.
“She studied his countenance like an inscription, and deciphered each rapt expression that crossed it, and stored them in her memory.” Change them to it.
“Each of them, in their turn, received the reward to which they were entitled.” This should be “Each of them in his turn received the reward to which he was entitled.”
No and not, like each and every, when they qualify a plural antecedent, or one consisting of two or more nouns, require a pronoun in the singular.
“No policeman, no employee, no citizen dared to lift their hand” Say, his hand.