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!
There is no penalty for building a city far away, though it does pose some hardships in practice:
It takes longer to move troops from one city from the other, which can be critical at times.
Moving airplanes and missiles between them requires special naval units, unless you have friendly cities in the middle.
Creating a trade route between them can take longer and be more expensive, see this related question on road maintenance cost - though this can be mitigated if they are both coastal cities, or close to other coastal cities, as you can read in this question which explains the trade network.
It takes longer to get the settler there in the first place - those are turns wasted on moving a fragile settler around instead of starting a city.
It takes longer to move workers and great people from one city to another, meaning more turns in transit (while still paying unit maintenance costs) and less turns doing useful stuff.
If you try to build a spaceship, and you build the parts in distant cities, it takes longer to move them back and they must be carefully guarded - spaceship parts are very lucrative units to destroy.
I think point (1) is the most important one. I've had enough instances where my distant cities came under siege and I could not reinforce them in time; I now try to expend slowly, though I admit it's sometimes very hard to resist building a city when coming upon a great location.
Point (3) is also very important but can usually be mitigated by just covering the long distances with the help of coastal cities.
Best Answer
A lot of wonders (if not all of them?) show up as a visible model around the city in which they are constructing / constructed. So if you see a pyramid around a city, you know the the great pyramid is built there.
It takes some practice to know what wonder is what, and it's definitely a very crude way; requires scanning all cities for things that are sometimes very hard to notice. But it does work, I use this method often to know which wonder is being constructed by other civilizations, to avoid working on the same wonder.
Notice that all wonders have different "under construction" and "completed" models.
EDIT: I've found this wonderful page which attempts to catalog all wonders with screenshots of their "under construction" and "completed" models. Still incomplete, but useful nonetheless.