1. I read somewhere that a city that doesn't produce Trade doesn't need a Library... but since roads produce Trade, every city needs a Library?
2. I read somewhere that cities should be close to one another... so if I know a city will never fully utilize all 21 of its fat-T squares, it's OK to have another city built so close that it will also use some of the same fat-T squares?
3. I read somewhere that cities should specialize and not build every building... so, in order to not build buildings, should I be building and immediately killing troops/missionaries? What I mean by that is if my city that is specializing in research buildings shouldn't also build a barracks, and I have to build something no matter what, what should I be building? (This question is related to before you can tell a city to focus on wealth or research but you have to build something.)

I'm trying to understand this game but... a factory limits the number of people a city can have, and the more people you have the more fat-T squares can be utilized... so why build population limiting factories?
I realize this is about balancing the benefits vs the costs as it relates to each individual city, I'm just having difficulty understanding how things work and how to get cities to a 20 population so I can utilize all the fat-T squares.

Right now, the only way I can compete with the computers while playing at chiefton level is to constantly cheat and use the worldbuilder. I'd like to learn how to play at noble level without cheating.

Instead of trying to explain things myself, I'll instead advise to read this excellent guide for beginners at a site dedicated to the Civilisation series:
https://forums.civfanatics.com/threads/beginner-help-the-basics.648469/

It is written by an expert player, but intended for beginners that wonder about the basics of the game.

Note that roads don't produce commerce (trade) in Civ 4. They did in previous Civs.

A library gives a bonus to science production, and since commerce is transformed into science, it makes sense to build libraries in commerce-heavy cities. But libraries also provide two scientist slots, so you might also want to build it in a food-heavy city (the food is necessary to support the scientist specialists). It depends on what you want to do with your city. You will very rarely work all 21 tales at the same time, so it's perfectly viable to share some tiles and switch them between cities as they need them. For example, switch a food-heavy tile between cities who need to grow their population. Buildings in Civ 4 don't have a maintenance cost. Instead, you need to consider the opportunity cost of not building something else that you need more. Training units when you don't have anything else to build is common, but be careful as units do have maintenance cost and building too many of them in the beginning of the game can tank your economy. I understand you mean that factories produce unhealthiness. As your cities grow, you will have to find new sources of healthiness (other buildings, resources, civics) to compensate this. The same goes for happiness/unhappiness. Read strategy guides and share your questions with the forum. Civfanatics is an excellent source for good quality articles, like the one Pangaea666 posted. If you prefer to watch a video tutorial, Sullla's Youtube series of Willem of the Dutch is another great way to learn the basics: https://www.youtube.com/watch?v=wm1CQxACekg

Thanks for your answers Pangea666 and ConsulCaesar.
One of the things I liked best about Civ's I and II was that I could build my civilization without ever having to be the aggressor (never declare war).
I took your advice, asked questions and watched/read gamplay suggestions. After it sunk in, I realized that one person' suggestion that 'all you need to build in a city is a Granary and troops to go capture more cities', meant the advice i was being given was about making war and not about how to build a civilization.
To me, 100 cities with just Granaries do not a Civilization make. Since the game mechanics allow for 100 cities with no buildings at all...
It seems to me that rather than be a Civ game with elements of war, Civ IV is a war game with elements of Civ..
I'm not into war games, so I uninstalled this game.

Neglecting your cities' infrastructure won't get you far in medium-high difficulty levels.

Anyway, if you don't like waging war, you should look into Space and (especially) Culture victory conditions.

This is not what people meant. Everybody builds a lot more than just a granary in each city. It is more to point out how important, crucial even, the granary is, because it allows your cities to grow faster (half the food is stored). But that doesn't mean one should build markets, banks and customs houses in all cities. They are more situational, and not all cities need them. One of the great things about this game is that we can customise cities to such a great extent. Your capital needs a lot of buildings, especially in a long game, but your newly captured ice-located city probably not.

You can certainly play Civ 4 without making war, or with minimum wars (only defensive ones for example). It makes things more limiting as you move up in difficulty level, because the AIs will there block you with their own cities, but it's doable. You can play with an Imperialistic civ for example, to help you with grabbing land. Or play at difficulties lower than Deity and Immortal where a de facto Always Peace game is more tricky. You can then try to make friends with other AIs and go for a UN or AP win, if that is feasible. Or maybe a Cultural win, which is typically done with few cities. Going to space is more tricky with only a handful of cities because you need to research the entire tech tree (a few exceptions aside), so it's easier with a big empire. I'd say a typical space game is to go to war early-ish if you can't claim enough land peacefully, and then tech up and cross your fingers the others don't declare war on you, by keeping good diplomatic relations.

This game can be played in a thousand ways, and that is what is so great about it. Unlike the later inferior imposters.