phineas and zac efron (i forgot the name of his character) take over 5 shots of liquor each in a bar/singing scene. they are never seen getting drunk though.
I would say no however, there is a scene where Barnum is kissed by Jenny lind without his consent however there is not mention of it being an assault or any flashbacks to it
A character is burned, while they are alive ; but if the question means Is someone burned to death" then No. They are rescued within about 30 seconds, and another character states they are still alive. They are then shown in a hospital bed, but no injuries are visible.
There is an intersex woman with a beard who is enlisted as one of P.T. Barnum's "freaks" to be laughed at. It's portrayed as uplifting, rather than degrading.