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!
The money gained from performing a diplomatic mission is 350 + (50 * <era number>)
where era number is just a serial number starting with 0 for the ancient era. In addition game pacing also affect this number: quick games modify this value by 0.67, epic by 1.5 and marathon by 3; or in other words:
Gold in Gold in Gold in Gold in
Era Quick Standard Epic Marathon
--------------------------------------------------
Ancient 234.5 350 525 1050
Classical 268 400 600 1200
Medieval 301.5 450 675 1350
Renaissance 335 500 750 1500
Industrial 368.5 550 825 1650
Modern 402 600 900 1800
Future 435.5 650 975 1950
I know of nothing else which can affect the gold amount.
Now, if I have a strong economy in the late game I typically gain another 100-200 gold per turn when in a golden age. That means that the only way in which a golden age is preferable to a diplomatic mission is when it lasts at least 3 to 6 turns. But usually by the time I get to the late game my golden age duration from great people is really low, and if you take into consideration that the influence gained is worth a few hundreds in itself, you'll see that a golden age looks less and less profitable in comparison. All this is especially true if you play in longer game speeds, which translate to higher gold.
Of course, golden age also nets production, so it's still a dilemma; but I do believe merchants are worth it from the financial aspect.
Best Answer
Yup.
The buildings are fundamentally equivalent in the game, which is why you need to build a Pyramid in every city (instead of a monument) if you're seeking to build the National Epic, and the same thing follows if you're building Steles instead.