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!
No, there are three possible ways to steal territory:
Culture Bomb with a Great Artist (Vanilla game only)
or
Build a Citadel with a Great General (Gods and Kings ExPack only, see PhysicalEd's answer)
The Traditional Method - just declare war and then capture whichever of their cities "owns" the tile in question.
The Diplomatic Method - convince the Civ in question to give you whichever of their cities "owns" the tile in question as part of a trade or peace agreement. You may find this method easy to achieve if using the Traditional method on some of their other cities first.
With both the Traditional and Diplomatic methods once you are in control of the relevant city you can either keep the city (and so the tile), or you could raze the city so all that land becomes free to be taken (by anyone!) and then purchase the tile/expand culturally from your nearest city.
The serious point here is that once a city unlocks a tile it belongs to that city for the rest of time, regardless of who owns the city, unless the city is destroyed in which case the tiles become "free" for other cities to take ownership.
The culture bomb (or building a citadel in the G&K expansion) is the only exception to this. Doing this captures all of the tiles around the tile where the effect is triggered, and this will take tiles away from any player or city state that had owned them (and I guess it gives them to the nearest city of the bombing player?).
Unlike Civ4, you cannot override a neighbour's borders by racking up a massive culture score in a bordering city.
Best Answer
According to this (fairly comprehensive) guide, a Culture Bomb will not convert a city. The author states that those long-range land grabs can still be useful even if they're outside of your cities' 3 tile radius utilization zone, to simply deny those resources to your enemy.