What are some ways I can reduce sound on this rental unit?
The best solution is to mask it.
Get a loud air cleaner or white noise generator and run it all night in your room. You'll sleep very well.
I like the air cleaner idea - solves two problems in one.
Another simple partial solution would be to buy a heavy "oriental" rug and hang it inside the door. This only works if the door opens outwards. The heavier the rug, the better. You should need to use screws or nails to hold it up. As much overlap as you can manage on the sides and top, and it should rest firmly on the floor.
Maybe two heavy rugs, back to back, for added mass, and so you don't have to look at the back of the rug when you are coming through the door.
You'll have to push the rug to the side whenever you enter or leave the room, which can be a pain if you are carrying a bowl of soup. And it will make your room look you are a big Rudolph Valentino fan. But it should absorb some of the noise coming through the door.
You may be able to rig up some cords to roll the rug up like a bamboo window shade, and let it down when you are going to sleep.
Here's another idea: Take the old door off it's hinges, and store it under your bed. Install a thick, heavy door with "weather stripping" around all four sides. When you move out, take off the new door and put back the original. Don't know about your lease, but this has worked in the past for me.
But, try the white noise approach first. It's simple, cheap, and you don't have to worry about the terms of your lease!
--- Chip