Can "wet" be used for liquids other than water?

I'd say no. "Wet" carries a connotation of non-disastrous. Someone covered in sulfuric acid is not going to be described as wet. Neither is someone who's just survived horrific flooding of a nearby river. We wouldn't even just use "wet" if it were important papers or a million-dollar-winning lottery ticket; we'd say something like "waterlogged beyond recognition." "Soaked," "drenched" (your suggestion), and the like, yes, but simply "wet" – no. That's what something/someone is that just got caught in the rain.

Two words: WET PAINT. Not all paint is water-based.

Also, we can consult a dictionary:

wet (adj.) moistened, covered, saturated, etc, with water or some other liquid
(from Collins, emphasis added)

It's very much context-dependent. Many things can be wet with various solutions or solvents during a manufacturing process. For example, this brings back memories from my days in the darkroom:

Color toners are applied in the darkroom after the final rinse. The toning bath is placed in a separate tray, and the wet print is submerged into the solution. > ref.

Most of the time, the liquid in question will be water-based, but that's because those are the liquids most of us deal with on a day-to-day basis. Yet I don't think this creates a restriction on the word's use, it just defines an area where we're most familiar.

^{English is not my native language, I might be wrong here}

To me, wet carries a connotation of coldness with it (more than e.g. moist). Even in the case of boiling water I would rather connect the word with its rapid cooling towards room temperature, rather than its constant-temperature state while in a heated kettle.

This would give rise to a definition as

Moistened with a fluid sufficiently volatile for evaporative cooling to cause a noticable drop in temperature of the covered object.

or perhaps, more in accordance with J.R.'s example,

Moistened with a fluid of similar volatility as water.

This is, as we know, true for water, also for alcohol or gasoline. It is not true for e.g. heavier oils, which seems to fit – one wouldn't normally describe lubricated items as wet (...except, of course, for one particular example, where however the lubricant is water-based...), it would constitute an unusual emphasis. Even less likely would you describe something as wet with sulphuric acid, or molasses, or molten butter, which differ from water in their much higher viscosity and again much lower volatility.

Chemists apply the term to water in an unusual way: water can make other liquids wet.

The other liquids are usually organic solvents. One of the reasons for these solvents is that the presence of water is a bad thing for whatever the chemist is trying to do. A wet solvent is usually a bad thing, and the term is used even if the amount of water is very small (usually it is employed when the water is a problem).

Example usage: http://www.ncbi.nlm.nih.gov/pubmed/11486383

Wetting has nothing to do with temperature, or the name of the liquid.

Wetting of a surface by any liquid is as someone noted above determined by a balance of forces, between those forces which cause the fluid to prefer to associate with itself (cohesion) and those which cause it to prefer to associate with a surface.

To wet means to adhere to a surface. That surface can be of a solid, or another liquid.

It depends on what sense of the word you're using.

As an adjective: Wet alone means it's soaked/covered in a liquid.

But it could look and feel wet independently: Something polished and/or covered in clear lacquer could look wet but feel dry. Clothes that are cold can feel wet without being wet or looking wet. Something covered in white-spirit is wet (covered in liquid that you can wring out), looks wet, but feels strangely dry.

As a verb: Wet means covering something with a liquid.

(Probably many other definitions too...)