This is an allusion to a classic catchphrase from 1950s:
Throw it up against the wall and see if it sticks, or sometimes
Throw 'em up against the wall and see which ones stick.
It means "Try it (or them) out and see if it is (or which ones are) effective".
It's one of a series of metaphors popularly supposed to be the natural idiom of the 'Mad men' the bright and irreverent young advertising men of Madison Avenue in New York. Their brainstorming sessions would come up with with ideas so unconventional that senior account men worried about whether they would even be acceptable to clients and to the public, to which the answer was "there's no way of knowing without trying - throw it up against the wall and see if it sticks".
Wikipedia has a brief article on an even more famous phrase of the same tenor:
Run it up the flagpole and see who salutes.
And there were many more; comedians and satirists competed to come up with new and ever more far-fetched variations. The website TV Tropes cites this as the first example of Cyclic National Fascination.
In the etymology you've posted, 'fun' is described as "to cheat, hoax" as early as the 1680s. 'Funny' is then attested at a later time as a modifier with a slightly modified sense emphasizing the humorous aspect of something like a hoax or joke. A particular 'cheat', 'trick', or 'hoax' could seem more like a good-natured laugh or more like a mean-spirited attack. You see this spectrum of meanings in the various definitions.
At it's core, 'funny' means 'unusual', 'unexpected', or 'odd'. All the definitions you've included in your questions are just different, more specific ways in which it is used to mean this.
Humorous
Things which are completely expected, known, and ordinary are generally not funny. Most humor involves misdirection and surprise, so it fits in with the ideas of unusual and unexpected, i.e. 'odd'.
Strange
I hope this is clear.
Dishonest
Things or people which are dishonest are not what they're expected or believed to be. Some uses of 'funny' in this sense include "funny money" and "funny business".
Unfriendly
I'm not familiar with this usage, but I'm not from the UK. The closest thing I can think of is "you're acting funny" when someone seems to be offended. This is said because the person is not acting in their usual manner, but instead in an offended manner. I suppose this could be extended to general unfriendliness fairly easily.
Ill
The important thing to note here is that 'funny' can mean "slightly ill". Feeling funny is between feeling healthy and feeling sick. You don't feel like you normally do, but you don't feel awful either. A simple stomach ache probably doesn't go beyond feeling funny, but the flu takes you all the way to sick.
A synonym for this sense of 'funny' would be 'quesy'.
Crazy
Crazy things are so unexpected and unusual that we often can't figure out why they would be that way.
Best Answer
From what I understand the first quote is from The Witcher.
Apparently in the original Polish they used pogarda, meaning a "mix of feelings of lack of respect, aversion and superiority over someone". (Reddit link.)
In English, then, they mean contempt as in "disdain and hatred for something". In this case, I understand that Geralt is going to be hated (by society? by himself?), and the lawyer thinks Geralt has to use that hatred against him as emotional fuel to find Ciri.
The second one is more straightforward. Contempt here more means "hate". It sounds like it means the speaker will get something, but only by doing something that will make them hate themselves.
Probably someone who's read The Witcher could make this clearer.