This is kinda why Civ can be either a science - or a philosophy. Some "high level players" have actually done the math and would be able to give you set rules with a mathematical certainty. The rest of us have developed our own principles, through trial and error - and some times habit, and have our personal philosophies. You will too, in time - unless you subscribe to some of the more "scientific" dogmas around. So for my personal views on this subject:
1. In a general since, what size cities are normal averaged at 1AD - 1000AD? Very generalized; AI trade can screw with the amount of happiness/health plus the overall quality of the map or even the need for growth itself. I just want guidelines on this, not instructions or set rules.
Depends on difficulty level and also starting location/map script. I'd say all cities should be at their happy cap, period. That is, be as large as they can without any wasteful red citizens. (They will only chew up the food but do no work.)
2. Is growth worth using an occasional farm on a normal cottage tile instead of a cottage? Question 5 asks the same question, but this is targeted towards farm vs. cottage rather than just about farms.
To cottage or not cottage is another question altogether. Early cottages are nice since they develop over time, but you can also build them later when Civics allow for a faster growth and new Techs promise even greater yields.
I'd say you pretty much have to go for Farms if your city doesn't have any

Resources (preferably a couple of them). Also note that you can get your cities to grow faster if you build all Farms first, and once at the happy cap, change them into Cottages. (Or go for specialists, which I tend to do early on to get my Great Scientist - or two - or three.)
3. Getting an early granary I hear is pretty crucial, is it as important as I hear. I know it sounds very newbish and I always prioritize it but I just want the scoop on why.
This has been a general rule through-out the entire Civilization series, so seasoned Civ players would tend to adhere to this dogma. Granaries double your growth rate, its as simple as that. Is that something you would want or are you already growing faster than you can improve the tiles around your city? There's no point in building a Granary if you could build something just as crucial, if you don't have the manpower (Workers) to improve the land or if you have to wait for a culture pop (or Tech) until you can hook up some luxuries (happy resources).
Before you use Slavery to hurry production (whip) you should always have a Granary first. This is an actual rule, because you would just be wasting

otherwise. In a worst case scenario - whip the Granary as your first sacrifice to the goods of blood and death and grain.
4. Is happy and health infrastructure an important priority fairly early as a preperation for massive growth? It seems you cannot always run HR and sometimes health is short on some maps. EX. Theaters/Aquaducts/Coliseum?
These are priorities and I would pretty much use the buildings that become available to me. Otherwise its easier to use HR to manage happiness. So, I'd go for Drama if I need the

boost. Theater is one of the more useful buildings though, so I would probably build these anyway - if I happen to have Drama early on. Same thing with Temples - if I have a religion and don't have HR, sure I'd build a Temple. Otherwise I could put it off and start building my army instead. One unit does the same thing as a Temple (minus the

, but the presence of the religion has already popped the borders).
Health usually isn't an issue early on - not in my games at least. But I only play on Noble anyway (for whatever reason). Other than that these are the two limitations to city growth. Given that enough

is available you're bound to reach either your happy or your health cap eventually, and then its time to stop growing. (You really don't have a choice.)
5. Farms: They confuse me; should you ever build them on a green tile (2+

) or just cottage it? This really is eating on me because it seems the benefeits of a cottage should be used on a green tile always. The only time I know I should build farms is on riverside plains (maybe watermills there too) and grain

bonuses.
Farms should go where they do they do the most good. That is, on (green) Grassland tiles. But you only build them if you need the

. If you already have

resources or have reached the happy or health cap, build Cottage instead.
You can also do a compromise and build Cottages on riverside Plains (+1

, +1

) and Farms on Grassland tiles to support these. If you feel unsure about this, just do the math yourself and see what would be the best solution for your situation. (Every citizen working a tile consumes 2

per turn, so with a total of 5 worked tiles, you need to produce 10

per turn. It doesn't matter how, with what improvement or on what tile. (Say you have a Farmed Wheat tile and four Plains Cottages, you're good.)