Alright, I started a game on Emporer and with a lucky Incan rush was able to take my opponent's capital, thereby eliminating all competition on my continent. The problem is, I decided to take his capital, which, after blowing all the money I got by conquering it, is costing me 6 gold a turn to maintain. My science slider is at 40% with no hope of increase anytime in the near future.

How far away does a city have to be before you'll burn the sucker? I knew something like this might happen, but it was in such an unbelivable starting position, and his workers had already improved the land...

I usually only raze cities that are "mis-placed" from my expansion plans. It sounds like you made the same mistake I seem to keep making. It's not so much the cost of the captured city that is killing you. It may be the neglect of commerce in your core cities that is the problem. One absoultly must get those cottages down early. It's going to be well over 50 turns before they are really worth anything and in the meantime there are other tiles that give better production. Using those other tiles may be a big mistake. I made this error in my current game, but thanks to a great merchant trade mission I was able to do some catch-up in the tech race, while also getting those cottages up to par. Don't worry so much about 6 maint on one city, worry more about why your capitol is only producing 4 gold surplus. (4 of course being an abstract number).

Is that all you have? Two cities?

At least halve of my Quechua rushes have been more trouble than they were worth. I don't take the capital city now; I just steal the worker, sit on their tiles, and kill and settlers trying to escape.

Early in the game, just raze the cities and pillage everything.

If you've implemented Caste System, you can juggle the city tiles and assign two merchants to cover the costs.

AI capitals always seem to be in good locations to me.

Quechua rush. Maybe Quechua are eating up maintenance?

I have already given up on that game, there are just too many obstacles to overcome, even restarting at 4000 BC I think it would be a long shot.

Problems:
Barbarians are crazy aggresive, and they have Axemen!
I selected Tropical as the climate for my planet. Aside from my first expansion, every other expansion is overcome by jungle.
The only happiness special resources on the island require calendar, or are underneath the jungle.
There is no source of copper near my starting location.
There is no source of stone or marble near my starting location.

Yeah! That's another problem with Quechua rushes. You can get yourself 50-60 points ahead of the other civs and so barbarians invade like crazy, to the point where you really don't feel like playing anymore.

Yeah! That's another problem with Quechua rushes. You can get yourself 50-60 points ahead of the other civs and so barbarians invade like crazy, to the point where you really don't feel like playing anymore.

It sounds like you know something about barbarians I don't. What thread would this info be in?

The higher your score, the more Barbarians will go for you, rather than the other leaders. It's a balancing factor with early leads in the game :)

I don't think it is absolute though, I am in game now where China and I both have similar scores, with China slightly ahead, but recently I have still been seeing my fair share of barbarians. Also, I imagine if your city cuts them off from their intended target they will still attack you anyway....