But bonafide belief does. Spirits are made of astral-stuff. I believe it went into that in Street Magic in the fluff.
This astral-stuff works like it does based on beliefs. A fire elemental has problems with water because metahumanity believes that water and fire are anti-thesis toward one another. Water puts out Fire and Fire boils Water. As such, Magic reflects that.
In SR2 I believe the Air and Earth Elementals had weaknesses toward each other as well.
Personally in the two cases mentioned...
I'd let the microwave thing do its trick because it's designed to agitate water and superheat it. It's not like you stand there looking at a water elemental and think / perceive unconsciously that it's an elemental made of astral-stuff that acts like water but isn't.
I'd rule the same for the liquid nitrogen. It'd work as it's supercold (not to mention non-flammable). It's out of the realm of mundane and into the realm of Tech so high it's magic. Now if you were spraying liquid Oxygen at a Fire Elemental, I'd have to ask if you had a death wish as the moment it turns back into a gas, it's HIGHLY flammable. Spraying something with a boiling point BELOW room temperature that happens to be extremely flammable while in a gaseous state at an object/critter that can burn through metal or concrete. Right. Good call there.
(Yes, I suppose you could extinguish a Fire elemental with liquid oxygen if you had enough to do it. Heck, you can do the same in real life with gasoline and a match. I don't think the elemental would be all that happy though.)
Now if you were taking a fire extinguisher and using it to put out the Fire Elemental, that makes sense and is a reasonable thing to try. I'd let it work. Basically, would you take that object to do something to correct a similar problem in real life? If so, then it should probably bypass ItNW. Fire extinguishers vs Fire spirits. Anti-coagulants vs Blood Spirits. Insecticides vs Insect spirits. Dessicants vs Water spirits. You get the idea.
Fire hose (spraying water) vs Water Elemental. Not so good.