There are sodium hydroxide cleaners for home use? Wow...There are. That's hilarious. Sodium hydroxide is one of chemicals people in another thread have been using to claim that solar cell production is more dangerous than nuclear waste.
It IS incredibly dangerous. Tons of things are sold to households that are incredibly dangerous. NaOH will turn your upper layers of skin into soap within a fraction of a second of getting it on you which isn't that bad if you know to rinse it heavily, but if not you could suffer some nasty burns. A dusty pellet in your eye could easily blind you, etc.
Please don't allow some petty grievances from some other thread to lead you into dangerously minimizing the hazard of caustic chemicals to the OP. That is not very responsible.
@OP:
whatever gas or vapour which was produced during the reaction?
Lye just turns fats into soap, which makes them more water soluble to wash away. Which is bad if the fats in question are part of your body, but not very dangerous once it's in your drain already and out of splash range, because it shouldn't produce any vapors as a result at all. It creates glycerol as a sideproduct, but that is not a gas at room temperature or pressure, and it is also quite water soluble.
If you have strong smelling gases, then
A) My first instinct would be that they are likely coming from the sewer or something else, perhaps now that you've opened up a hole for them by melting through other stuff with the lye. Which shouldn't be happening no matter WHAT, even with a dry U-bend, and implies that the plumbing is installed all incorrectly in the dorm, with different drainage lines illegally crossing one another without proper traps and things. If so, you should demand a plumber and inspection for your own health. Not trying to fix it yourself.
B) HOWEVER, you seem insistent that it's not a sewer or anything, but is the NaOH itself. This is odd, but I guess possible. It might be that some other tenant / student / whatever has recently poured some weird shit in the drain that is having some unforeseen reactions with a strong base. Like I don't know, if for some bizarre reason, somebody dumped a bunch of aluminum bits into your sink (or maybe some really bad cosmetics...), then NaOH could react to form fairly dangerous and flammable hydrogen gas.
This shouldn't normally happen, but people in dorms are stupid. In any case, you have no way of knowing exactly what it is or how toxic it may be, so just turn the sink on a bit (unless it's backed up), open the windows, turn on fans, and get the hell out and call maintenance to evaluate. Not your job.
I had tried disassembling first, but despite initial successes, I was unable to do it completely. I suspect too much limescale buildup has petrified the whole thing.
I don't think in this case you should be trying to do this yourself, but for general FYI: limescale generally reacts quite well with vinegar (similar to why marble statues and stuff melt in acid rain), which crumbles/dissolves it and makes quite harmless byproducts. So harmless, in fact, that oral antacids are made out of limestone and almost the same reaction happens in your stomach when you take them, to no ill effect.