I wanted to see why the smell in the bathroom was not going away, and I opened the fan grille and noticed that the bathroom fan has no duct.

Inside the grille is just a small box with a fan blowing upwards into it, but there is no way for the smell to escape by means of a tube to the roof.

Is this bad? Could it be the reason my apartment has a generally stuffy atmosphere, even with the AC running?

Are you saying that the box above the fan has no openings at all? If that's the case, it is against code. You should point this out to your landlord and have it corrected.

It would not, though, cause a "general stuffy atmosphere" in your apartment.

Hmmm. That's what I thought. The problem is that there would be no way to get a pipe going from the fan to outside by means of the roof.

Maybe not stuffiness, but do you think it could be the cause of more stale air (not as fresh)?

No. It's likely that you cannot "feel" any difference in the atmosphere of the apartment with the fan basically non-existant... but if you did, what you would feel is that the apartment is more humid.

If it feels stale, then it's probably a circulation issue. Check your furnace filter. If it's clogged up, that could contribute to a "stale" feeling.

As for the fan exhaust, if it can't go to the roof, then maybe you could run it between joists and out the side of the building.

... but if you are renting, then it's the landlord's responsibility, not yours. Depending on how important it is to you, you could push it as a Code Violation and he would be required to meet code... which probably would mean either making it operational or adding a window to your bathroom.

Well, it's not the filter since I change that monthly and I just put in a new one.

It really isn't too stale, just that othe houses I have been to have what seems like "fresher" air. I don't know if that is because they are bigger or what.

It doesn't really feel humid at all, because the AC stays on for a few hours at a time and does a good job of removing moisture.

I noticed that the stove fan vent do not go outside either, but rather exahust air in a slot above the stove light into the living room. This is the reason it heats up the air a lot when I cook a big dinner, but it probably doesn't contribute to the air quality.

The bathroom is completely indoors and getting windows would be impossible (if possible, that would never happen).

Could it be that the buildings were built (early 80s) before such codes were in place, or just that they didn't feel it was important to take the effort into doing it?

The vents over the stove are possibily ductless vents and if they are they are used to collect the grease that come off the food as you are cooking. There is is filter in this type vent that should be washed every few time that you use you cook stove.
As far as the bathroom is concerned, is this so called vent also part of the lighting system or does it serve as a heat-a-light for the bathroom? If this is the case the only purpose of the fan is to keep the lights cool, they get very hot in that box and can create problems lile over heated sheetrock making it very brittle and it cracks. Also the insulation on the wires will become brittle and could break off.
If this information does not fit your situation I am sorry for bothering you.