The right option to the question: <em>¿Qué clases de animales existen en este planeta?</em> is:
<em>En este planeta </em><em><u>existen</u></em><em> toda clase de animales.</em>
Explanation:
The translation of the sentence is:
On this planet <u>exist</u> all kinds of animals.
The verb "existir" in Present Tense has the next conjugations depending of the personal pronouns:
<em>Yo </em><em>existo</em><em>.</em>
<em>Tú </em><em>existes</em><em>.</em>
<em>Él </em><em>existe</em><em>.</em>
<em>Ella </em><em>existe</em><em>.</em>
<em>Ello </em><em>existe</em><em>.</em>
<em>Nosotros </em><em>existimos</em><em>.</em>
<em>Ellos </em><u><em>existen</em></u><em>.</em>
The noun within the sentence, that is, who performs the action, are "all kinds of animals" that could be replaced by the personal pronoun <em>"ellos"</em> or "they" in English, so, the right conjugated form of the verb <em>"existir"</em> that applies is <em><u>"existen"</u></em>.