Choose the correct word.
"His or her" is correct. "Everyone" is a trick singular pronoun. Since it is singular, the plural "their" is incorrect. The single gender "his" or "her" also is incorrect, since we know from the names in the sentence that the group of "everyone" is made up of both females and males.
Everyone took his or her umbrella except Maude and Oscar.
"Is" is correct. "Each" and "one" are trick singulars which require a verb in its singular form.
"Hey," he cried, "each one of you bullies is despicable!"
"Cares" is correct. "Nobody" is a trick singular that requires its regular, present-tense verb to end in -s or -es.
An old, popular song started with the line, "I ain't got nobody, and nobody cares for me."
"Is" is correct. "Someone" is the trick singular that requires a singular verb.
Someone is going to pay for this!
"His or her" is correct. "No one" is a trick singular which requires a singular pronoun. While "its" is singular, "the panel" is made up of people and not things, so you need to use "his or her."
No one on the panel decided to offer his or her real opinions.