I just moved to a new, old apartment. At first I did not notice anything, however I am now smelling an old, greasy, cigarette sort of smell in the kitchen near the stove and stove vent. I understand not everything being perfectly clean. However, after taking apart the entire vent yesterday, cleaning it with germicide, degreaser, lemons, limes, and more, and fanning the entire apartment, the smell has returned. I stained a coffee table yesterday, which may be the source of my cough, however I have noticed I am coughing much more often when I'm in the kitchen. A similar sort of smell appears when I run the fan in the bathroom upstairs.
My main questions:
It sounds like the range hood duct wasn't quite airtight, and some grease has escaped into the guts of the structure it was running through. This type of IAQ problem is basically a nuisance -- it's easy enough to fix by removing the culprit grease buildup from wherever it is, although finding it may be a hassle and may require management to dispatch someone to help by getting into the wall/ceiling cavities (which is where this grease buildup most likely is, either that or inside the motor). Also, seal the range hood duct with some high-temp foil tape while you're at it, and make sure there aren't random greasy buildups behind the stove or wherever.
I wouldn't focus on IAQ monitoring -- the best approach to air quality problems isn't to monitor, lab test, or even ventilate, it's to follow your nose to the source and eradicate it. Quoting Max von Pettenkofer:
If there is a pile of manure in a space, do not try to remove the odor by ventilation. Remove the pile of manure.
For more information, I would suggest reading the work of folks who have more experience with this than I, namely BSI-017 on why IAQ problems aren't going to be fixed with fancy tests, and BSI-070 on why venting isn't an answer to everything.