Does the "Diplomacy" skill and the artifacts associated with it only affect the surrender price, as stated in the description, or do they also affect the likelihood of monsters remaining loyal? And when the game says that monsters want to join, is that just random, or does diplomacy only influence the situations where "monsters want to join for a certain amount"?

Diplomatic artifacts, according to the game manual, only affect the cost of surrender if your hero wants to escape the battlefield while preserving all (remaining) troops. They have no effect on monsters. :)
However, the Diplomacy skill always plays a role when monsters decide whether to join you, and whether they will ask for money to join is a separate issue.
Regardless, read the game manual (section 5.6: Diplomacy and its specific applications): it describes all of this in detail and in an accessible way, including with an example.