Was Lennie's death considered euthanasia, some form of death penalty, or murder?

This is a great question.  In some ways we can argue that George's act of taking Lennie's life was a form of euthanasia. Euthanasia comes from the Greek words, "good" and "death."

When George found out what Lennie had done, he knew immediately what would happen.  He knew that the men would come after him with full force.  We need to keep in mind that Curley was the boss's son.  So, all the resources of the ranch would come after Lennie.

To be more accurate, George knew that the men would catch Lennie.  When they did, they would not only kill him, but also cruelly kill him.  His death would be prolonged and painful.  Suffering would be multiplied.  In light of these certainties, George did what he thought was most merciful.  He took his friend's life in the most painless way.  He even gave his vision of the land that they would have together.  In other words, George did his best to ease his friend's death.  Literally, George gave Lennie a good death.