Learn Dutch: Niemand mag de gijzelaar vermoorden. - No one may murder the hostage.

This sentence is actually a contrast to the object control verbs you are learning. It uses the modal verb mogen (to be allowed to). Modal verbs like mogen, willen, and kunnen always take a bare infinitive without te. The verbs in this lesson's main topic (like advising or forcing) require te or om te.

In the other sentences from this lesson, the object performs the action of the infinitive. Is that the case here with de gijzelaar?

No, this sentence works differently. In an object control sentence like Ik dwing hem te luisteren (I force him to listen), the object (hem) is the one doing the listening. Here, niemand is the subject who is not allowed to act, and de gijzelaar is simply the direct object receiving the action (being murdered).

What is the difference between moorden and vermoorden?

Moorden refers to the general activity of killing, often used when no specific target is mentioned. Adding the prefix ver- turns it into vermoorden, which specifically means to murder a certain person or thing. Because the sentence specifies a direct target (de gijzelaar), we need the transitive verb vermoorden.

Could mag in this sentence mean probability, like "No one might murder the hostage"?