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!
Here is a list of all the effects from happiness level that I could find. To answer your specific question, most of the effects care only if your civilization is happy or unhappy, by one :) or :(. (I believe zero :) is fine.) But there are three effects that care about the amount your civilization is happy or unhappy:
- VERY unhappy occurs when you hit -10 happiness. (Thank you @Oak)
- Golden Age points accumulate (or deplete) equal to the amount your civilization is happy (or unhappy)
- Piety->Mandate of Heaven uses your exact excess happyness to calculate how much culture you get.
Full list of effects:
Happy
- Growth: Cities grow normally.
- Golden Ages: If you have excess happiness, each turn it accumulates into Golden Age points, and when you have enough, your civilization enters a Golden Age. After the golden age the process starts again, although I believe you need more Golden Age points for each subsequent Golden Age.
Unhappy
- Growth: Cities grow at 1/4 speed when your population is unhappy.
- Golden Ages: The amout your civilization is unhappy is subtracted from the Golden Age points.
VERY unhappy
- Growth: Cities do not grow.
- Units: Settlers may not be trained.
- Production: Cities generate -50% production.
- Unit are less effective in combat. They get -33% combat strength.
VERY VERY unhappy (-20 or lower)
- Rebels will spawn in your border and attack your city
Social Policies
- Piety->Mandate of Heaven allows you to add 50% of your excess happiness to your culture each turn.
- Rationalism->Sovereignty boosts science +15% when your empire is happy.
- There are numerous social policies that affect the happiness level, but these are the ones that get effect from the happiness level.
Sources:
- General info: mousing over the happiness listing on the top bar, left side, during various happy states.
- Golden age info: mousing over the golden age display on the top bar, left side, during various happy states.
- Combat info: Comparing happy (normal) combat strength to VERY unhappy combat strength, by mousing over an enemy unit. The VERY unhappy combat modifier is clearly listed in the calculations.
- Production info: mousing over a city's production listing in the top left when VERY unhappy.
- Social Policy info: The social policy screen, accessible by clicking on the culture listing on the top bar, left side.
Best Answer
Actually I think in some cases it still might be not such a bad idea to do that. Especially if you didn't invest too much in that policy and you are not going for cultural victory anyway. Some early game policies matter much less in the late game. For example you can use a bit of Liberty for early expansion and then abandon it for Autocracy.