1. The landlord refinished the cast iron bathtub with aircraft paint. It looks good, almost like ceramic refinishing. Would there be any health hazards from soaking in the bathtub?
Old-style paint with lead in it was dangerous if ingested or breathed in. Aircraft paint may have chrome in it, which again is a heavy metal and isn't good for you to breathe, so painters using it need appropriate PPE. I don't think that it can come off and hurt you unless you find a way to ingest or inhale it. From what I've read, though, the chrome-bearing aircraft paints will cause rust in iron and steel, so the chances are that your landlord used a "green" aircraft paint without heavy metals in it.
2. I have a portable AC that has to run a duct to the window. The duct gets hot. I went to the hardware store to look for insulation. All the fiberglass insulation products have warnings that scare me: to install it, I have to wear an N95 mask, goggles, long-sleeves, and then launder the clothing separately, and rising the washing machine after. I just need to cut the insulation to length and wrap it around the duct. What kind of risks am I facing? Also, do I need to somehow cover up the ends?
Fiberglass insulation has two main hazards that I'm aware of: inhaling it can cause all kinds of lung issues including silicosis, so don't do that. It can also cause skin irritation. I wouldn't be too concerned about the application -- just follow the directions and use the right PPE -- but I wouldn't allow fiberglass insulation in my house except protected, like inside a wall. In other words, I would not apply fiberglass insulation to an exposed duct unless I had some plan to protect the insulation from being disturbed.