Although bwarner's answer was helpful as always, no one verified beyond "I think so" that puppet cities are affected by each of the civilization wide effects, so I did some testing and:
Yes, puppet cities are affected normally by all civilization-wide effects.
I can't verify that there are no exceptions, but I verified at least one of each of the things I was concerned about...
The Testing
Civilization Special Trait
I loaded up France on easy, and went about conquering. Japan fell, giving me Kyoto to play with. As soon as I conquered the city and took it as a puppet, it was producing 2 culture. This matches perfectly with France's ability, so CHECK!
More Test Setup
Having acquired my victim, I setup the test: social policy ready, golden-age ready, and wonder ready all on the same turn! (Save file available upon request, whenever I get around to it after that.)
Before
Social Policy
I enacted Liberty->Republic, which gives +1 production/city. That matches, so CHECK!
Golden Age
I already had culture working for me, so I started the golden age by enacting the social policy Piety->Reformation. Clearly shows an increase in both gold and production, CHECK!
World Wonder
If you take a look at all of my cities from the same turn, you can see Orleans is about to finish the Sistine Chapel, which gives +33% culture/city. I went to the next turn, and without any growth or buildings completing, the after shows increased culture on Kyoto. (Lyon also started with 7 culture and ended with 9 as well, so the rounding definitely matches.) CHECK!
Everything checks out. I suppose puppets really are just normal cities that you don't control. I would still love to hear if anyone finds any exceptions!
Normally a City-State will just raze any city that it managed to capture.
This could make giving a City-State lots of units a handy way to destroy a city or two in the vicinity of the City-States (remember, they don't roam far from their borders) without getting the world mad at you; you just have to ensure the CS stays at war with the target player until the desired target is destroyed.
However, for any city that it captures that cannot be razed - that is the original capital cities of the main Civ players and any other City-States - the capturing City-State simply takes the city as a puppet, leading to the odd scenario where a City-State player controls more than one city!
What they do not do is liberate the target if it could - for example, in my current game, Venice has just captured Stockholm, which was being held by the Aztecs. They didn't liberate Stockholm, just took it as a puppet, and so Venice now controls both cities (and the Stockholm player is still "out" of the game).
Best Answer
Take the trade route between Te-Moak and Milan. It is earning me 19.8 gold, and it is earning Milan 1 gold.
If you hover the city names you can see an exact breakdown of how this is earned and how it is divided between the two parties.