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!
Sounds like you were unfortunate to not spot the issue earlier; probably the best thing you could have done at that stage would be to fortify your borders, hope to deflect the first wave (when it eventually comes) and then counterattack, charge in and capture and hold a few of their cities (thus making them weaker). Although if you were aiming for a cultural victory you would need to avoid annexing any, otherwise this will make your social policies cost more.
In my experience Montezuma and Augustus are, by default, quite aggressive when played by the AI; if you find an AI playing either near to you, I feel it is best to plan ahead to stop them expanding too much and have sufficient defence on that border - or instead try to make sure you get friendly with them so they're on your side instead.
Also, I believe entering in to a pact of Secrecy with other Civ's against a threatening Civ earlier in the game would have helped prevent the other Civ's becoming allies with the threat later on, by ensuring they have a "bad" relationship with the threat. But I don't have hard evidence for this.
Best Answer
When you capture someone's capital, their capital is temporarily relocated to another city. But it is the original capital that is important for the victory condition. You are not allowed to destroy it, so you need to be able to hold on to it until you are the only one that still controls your original capital.