First thing’s first, Shakespeare took out the first major character (other than the original king). The murder of Polonius was intense. I read what happened and I laughed, yet at the same time I was deeply disturbed. As I have said, this is not pretending, he’s really mad. If I were in his shoes, I would have revealed the person and knocked him out. That is completely different than blind murder. His excuse is “The body is with the king, but the king is not with the body” (IV.ii.25–26). I must say, this is a good reason, but the act of murder is HUGE. Hamlet doesn’t understand, two wrongs don’t make a right, and ten wrongs is just insane. In act 3 he nearly murdered Claudius, teased his mom, ridiculed Ophelia, and he stabbed Polonius. The murder advances the reader’s opinion of Hamlet’s character, because now Hamlet and Claudius are murderers. Hamlet replicated the same problem for Ophelia and Laertes. It’s not fair to do the project your problems on others, which loses a lot of people’s sympathy for Hamlet.
Claudius was not cool after the play. He totally took the bait, so now he knows to get rid of Hamlet. After the murder of Polonius, it’s time to execute Hamlet in England. Claudius knows that if he stayed in that room, he would have been in serious danger. In Claudius’ shoes, I would do the same. Letting a psycho who knows about your murder and wants to kill you go on the loose is a bad idea. No matter what, Claudius must get rid of Hamlet. It also needs to be done in England, because everybody at home loves Hamlet.
Poor Ophelia has gone mad, as seen in scene V. She slips in and calls for Hamlet, obviously still loving Hamlet. Also she is upset about her father’s death by her love Hamlet. Naturally she is insane. She has gone insane because of what society did to her. Laertes and Polonius told her not to sleep with Hamlet. Society has told her to be a good little girl when she wanted Hamlet. Then Hamlet went insane and killed her father. These events took a toll on Ophelia through collateral damage. Now she slipped into madness, singing and saying things like “They say the owl was a baker’s daughter” (127), coming from the legend in the bible. Personally I find much sympathy for Ophelia’s tragic case of insanity, because Claudius and Hamlet are murderers and she’s innocent.

You are so right about the mess that Hamlet creates impacting our ideas about him. Can we think of Hamlet as heroic after he wreaks such havoc of the lives of others? I am glad you grapple with this question.
Ophelia indeed suffers “collateral damage.” I am with you in feeling genuine sympathy for this tender personality unable to withstand the “slings and arrows of outrageous fortune.”