Your wrote a lot of side questions, but I will answer what seems to be the main one:
Can a player rejoin a Civilization 5 multiplayer game after being disconnected?
In theory, yes. I've personally done this. Try reconnecting by finding your friend on Steam, right clicking on them, and selecting the "join game" menu option, it will take you right back to the game.
However, its quite possible for whatever caused the disconnect in the first place to prevent you from reconnecting. I've personally experienced this as well, due to my internet connection simply dropping. If this is the case, that problem needs to be fixed before you can reconnect.
If you are having problems with a bad connection, there is one more thing I'd suggest you consider. As of now, in multiplayer Civilization 5, you can't just save the game. The only saves you have are the autosaves. So, I'd recommend changing your settings so that you autosave every turn. Then, at worst, you can reload the last save from one turn ago.
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!
Best Answer
One City Challenge would not be affected by No City Razing. When you take a city in OCC, the city is instantly destroyed. This can even destroy otherwise unrazeable cities such as Capitals and City States.
The only effect No Razing would have would be preventing the AI from razing other AI players' cities.