Hamlet the Avenger

If Hamlet never saw the Ghost, would he still avenge his father's death?
I believe he still would because he had built up anger in him, all it took was a ghost (which could've been a figment of his imagination) to push him towards avenging. If the ghost was a figment, then yes he would've avenged no matter what because his subconscious would've imagined the ghost so that he would have a reason to push himself to avenge. I personally believe that the ghost is a figment, but my theory could easily be diminished since others are able to see the ghost. Perhaps the ghost was a real ghost and everyone (except the Queen) could see it, but only Hamlet could hear it because he was making up the words in his head. Or maybe the ghost is the devil and came to destroy the city because of all the corruption that was happening, which would make sense why everyone dies. To me, this theory makes the most sense. The ghost is a devil but everyone sees it as a ghost. The devil talks to Hamlet because he knows if he acts to be Hamlet's dad, then Hamlet will definitely avenge Claudius. The devil/ghost knows that Hamlet will act as a catalyst towards destroying their kingdom.
Is Hamlet actually mad? Are other characters mad?
I believe that anyone who can see the ghost is considered 'mad', so perhaps the Queen is the only sane one. But I personally believe that Hamlet isn't 'mad' because he's just angry. Since he didn't see a therapist, he has anger management issues and built up anger that he's ready to kill because he hasn't developed coping skills to control his anger and frustration. I understand why people argue that he's mad, because he acts incredibly irrational at times, but I believe that he is just so angry that he doesn't know what to do with himself. And in Alice and Wonderland, the Mad Hatter says that everyone's a little mad, so that would make sense in this story.