Debt? No (except, IIRC, in Chieftain). You can have a negative income and not have any problems, though. But once you are at 0 Gold and still losing money, you start losing Improvements or Units.
Yes, you can simply spend less.
Yes, you can simply spend less.
You can't raise taxes in Civ3.
You forget the "give me the cash or my cavalery will kill you, Gandhi!"-tax, which can be raised.
Hmmm nope, the sliders can decide on how much of your overall income you spend (science-happiness), but you cannot generate more income by simply moving the sliders.