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!
I've been thinking about it myself, and I agree that it's almost always a better option to let the city stay as a puppet for a while before annexing it.
One major drawback here is not only that you cannot control what puppets produce, but it's that puppets never produce units. If you're at war, for example, the ability to produce units is critical, and sometimes you just need the extra city right now. Still, considering that conquered cities lose half their population and most of their buildings, their production is usually so low that making them a puppet is probably still the best choice.
Regarding the edit: I see no reason for annexing a city before the riots are done. You can't do anything during riots anyway, so it's best to not lose the extra happiness for these few turns.
However, now WillfulWizard's answer becomes more important. If you intend to annex it soon but you want to keep it a puppet for just 2 to 5 turns, there's a good chance you might forget - particularly if you're in the middle of a war and every turn is hectic and eventful. If you intend to annex it quickly anyway, maybe it's best to just lose the few points of happiness from having the city annexed during its riots, than to forget about it completely.
Best Answer
Just left click it =)