Beorn had a few instructions for the party.
Gandalf tells Beorn the story of the journey from the start to the point that takes them to Beorn's house. Beorn is intrigued and entertained and so, as the group are no threat to him, he allows them to to stay in his house for the night. While they sleep, Beorn travels to discover if the tale he has been told is true. It is and as they have killed the Great Goblin (Beorn hates goblins) he agrees to give the group supplies and ponies to take them to Mirkwood.
They did not take Gandalf's and Beorn's advice to STAY ON THE PATH while traveling through Mirkwood.
We know Bilbo, the hobbit, was well guided and well guarded on his way home because Gandalf, the wizard, accompanied Bilbo all the way home to his hobbit-hole, while Beorn, the shape-shifter, accompanied him just until they approached his house in Mirkwood. Also because Beorn gives very good advice about Mirkwood.
Gandalf advises the group never to stray from the path while traveling through Mirkwood. He warns them about the dangers of the forest, including the disorienting nature of its surroundings and the various threats lurking within. Staying on the path is crucial for their safety and to avoid getting lost in the dark and treacherous woods.
They enter in pairs because Gandalf tells them to do it. They don't want to go in all at once, so Beorn will think they were trying to kill him The dwarves must enter Beorn's house in pairs so that, as Gandalf tells their tale to Beorn, each entrance does not seem very imposing and Beorn's interest in the story is built up. Then the group will be able to stay with Beorn.
He retains Beorn's interest by introducing a few dwarves at a time, while he is telling his story.
Gandalf presents the dwarves in a staggered fashion while telling a story to keep Beorn's interest so he would not send them away.Gandalf wanted to keep Beorn fascinated in his story so he wouldn't notice the large number of dwarves Gandalf brought.
Beorn revealed the head of the goblin stuck outside the gate while the skin of the wolf was nailed to a tree just beyond
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They're only mentioned once that I recall in LOTR, by Gandalf while he recounts a history -- they aren't in the narrative that I recall, although I think Silmarilion may touch on them. Typically, skin changers or shape-shifters are those who can at will adopt the apprearance of other persons, and in more extreme cases, animals. Depending on which legend you're talking about, they may adopt the abilities and characteristics of whatever they change into -- or they may not. JRRT doesn't go into this. JRRT's descriptions in Silmarillion of the Eldar days also included vampires, werewolves, and quite a few more typical monsters. The most famous skin-changer is Beorn, who turns from man to bear when he needed to. He is a key character in The Hobbit, providing food and shelter to the party for a few days, as well as being one heroes of The Battle of the Five Armies.
A goblin's head that was stuck outside the gate and the skin of a warg nailed to a tree.
The Necromancer, who actually was the Dark Lord Sauron in hiding before he reentered Mordor. While this did help the health of Mirkwood some, its main effect was to cause Sauron to actually return to Mordor and call up his servants the Nazgul.