Beaten by Grille.....the shame, the shame....
Picking on one statement you made wW (welcome, too, belatedly)
Originally posted by wildWolverine
How exactly does "Science" at each city work? Meaning, how do scientists affect my research rate? Does each city contribute a certain research value to my civ, and then my overall rate is determined by a gestalt? My apologies for being so inept at Civ3...
I'm going to assume from your question about scientists that you're making the common error of using scietists and taxmen (which the manual kind of encourages, but it's not the most helpful thing you get with the game, unless you have a table with a wobbly leg

)
Never, ever, use specialists (scientists, taxmen or entertainers) if there is a tile the citizen can be used on, unless you are forced by extreme city unhappiness. (OK, 'never ever' is a bit strong, there are some minor exceptions, but not that many)
Specialists contribute NO food, NO shields and only ONE of whatever they specialise in. Even on a relatively poor tile you'll get one of at least a couple of the basics. Especially once you improve the tiles - even a desert will give one of each when irrigated and roaded.
Also, improvements don't affect specialist output. So let's assume you have 50% set for research, and you can either work a 2 commerce tile or make the citizen a scientist. Either way you get one science beaker, but the working citizen also generates an extra commerce (might go to treasury, might go to entertainment, depends on the slider settings) plus maybe food and shields.
If you have a library (+50% science) and a university (+50%) that will convert the working citizen's 1 beaker to 2 beakers. The scientist still only gives one beaker. Clear advantage to the working citizen.
To answer your actual question (sorry to be wordy there):
Each city independently totals all the commerce produced by its citizens.
Each city then independently splits that commerce base into research, entertainment and treasury values, based on the rates you set on the science and entertainment sliders (Treasury gets the balance, of course).
Each city then considers the improvements in the city and factors the base science or treeasury output by the appropriate percentage. I *think* it rounds up but it might not.
The total across your whole civ is then determined by adding all the cities together. This is then added to your treasury, or to the current research beaker total (which you can't see, but is implies by the steady reduction in the turns to the next tech)
This per-city calculation means some odd and/or annoying things happen.
If you have 2 cities with 2 commerce each, you CANNOT set the slider to get 3 beakers for science. You can only get 0 (0 in both cities), 2 (1 in each) or 4 (2 in each). If you adjust the sliders in the early game and look how the science output varies you'll see it's not a smooth variation with the slider, as each city jumps around due to rounding off. Later in the game it's not so noticeable, since you have more cities and higher commerce totals, so the rounding is a lower effect.
The fact that you set a single civ-wide percentage means you sometimes have to compromise. If a core city is perfectly happy (due to lots of improvements - temples etc - but an outlying city is rioting, you may want to increase the entertainment tax to make the rioting city happy. But that creates 'excess' happiness in the core, where it wasn't needed. This is one of the cases where local entertainers in the rioting city IS appropriate.