You guys were super helpful, and gave me the info I needed to call. And I called.
According to the housing department, everything about the situation was illegal, and they were quick to jump to. My landlord was forced to install a brand new radiator in my room. There was in fact a leaky valve in the radiator, which was causing it to drop to such a crazy low temperature. And I went to a doctor and started treatment for my various illnesses, which all appear to be mold related. Right now we're just symptom managing, I won't be able to 'heal' until I'm out of the house completely.
And now it's apartment hunting time :/ I've seen four or five places this week, all of whom told me they'd 'get back to me'. I feel like I'm continually failing interviews, haha. Finding an apartment in New York: unlike anywhere else. I keep going in expecting to sign a check and instead being interviewed and being all "how do I make people like me??". But without you guys, I'd still be shaking in the cold. And, of course, it snowed all day in New York :P