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!
So based on Fabian's answer here combined with Oak's experience mentioned in the comments for this question, it sounds like razing a city will cause all tiles that were acquired by that city to become neutral, as well as all tiles in the immediate ring around that city, even if those tiles were not initially acquired by the city being razed (beacuse as Fabian pointed out, building/capturing a city will automatically transfer ownership of tiles in that inner ring that belong to other cities of the same player to the "new" city). Any tiles in the inner ring that do not belong to the player owning the city should not become neutral.
Unfortunately, I am not aware of any way to determine ahead of time which city initially acquired a tile and thus "owns" it for purposes such as this.
Best Answer
To work a tile means you have a citizen on it, not a worker unit. To see which tiles are being worked, click on the city and enable manual citizen management. each tile within your borders will have a circle that is either gray (not being worked) or green (is being worked).