1- If a resource (say Ivory from elephants) is near a city (within culture boundaries) and I improve it, will the city benefit? or I need a road to the city?
You need to connect it via road to receive the resource bonus (ie ivory, silver etc). Even without a road, you can still work the tile and get food, production and commerce etc
2- If this same resource is touching a river, and that same river passes by the city (see picture) will this make it connected resource? Or I still need to build road to the city from the resource?
If the square has access to "fresh water", then it is touching the river and will get access to any resources on the square just as if the river were a road.
If a given square doesn't have access to the river, then you don't need to build a road all the way to the city, you just need to connect the road to the river
Except for reducing movement penalties, rivers are in all ways equivalent to roads. You can interchange any combination of roads and rivers to connect your cities for purposes of trade routes, resource connections etc
3- What is the math behind city growth?
Don't know. It's been mentioned in the forums somewhere before, but without search, I can't find it.
It says will grow in 7 turns, with 9-6 food. Is this like 6 food to support workers, but 9 food produced by city, so excess is 3 food goes to grow?
That's sort of what it means. Each unit of population eats 2 points of food. Food income is also reduced by unhealthiness in excess of your health rating. Unhealthiness is added by certain buildings and terrain, as is health. you can view a good summary in this thread
http://forums.civfanatics.com/showthread.php?t=145249
As for how much excess food is needed to grow? Beats me. It's in here somewhere, but with no search, I can't find the thread. Suffice to say, the more excess food, the faster it grows.
4- If a river connects two cities, do I still need to build a road between them? Or it is automatic trade using the river? How about coastal city?
You don't need to build a road, no. It counts as a road for all purposes except movement reduction, so that means the cities are connected.
As for coastal tiles, once you develop Sailing (IIRC) coastal tiles count for trade routes etc. This means that two coastal cities on the same continent will count as connected. More than that, it means that any city connected to a river that is in turn connected to a coastal tile can connect to any other coastal city. More than that, it means that any city connected to a road connected to a river connected to a coastal tile can connect to a coastal city (or another city connected by any combination of the above).
I believe, though am not sure, that a resource sitting on a coastal tile will connect to a coastal city with requiring rivers or roads as well (once you develop sailing?)
Finally, compass (IIRC) lets you use ocean tiles for trade, letting you make connections between continents
5- What affect how much gold per turn I produce? How to improve that immediately? Build less units?
There are several answers to this. Firstly, the more units you have, the more money it costs. They generally won't be your biggest concern. The bigger concern is generally the amount of cities you have, and their distance from your capital. You can see this figure as your maintenence figure in each city. Building courthouses helps reduce the maintenance cost. Maintenance is a big factor in limiting how fast and how big you can expand in the game. It can literally send you bankrupt if you expand too fast for your treasury to keep up.
There are also buildings that increase the amount of income your city produces. These include banks, markets, grocers etc. The thing to realise is that these buildings increase your commerce by a percentage. So the more money the city was earning, the more benefit you will get from these buildings.
Finally, there are certain tiles and improvements that will increase each cities commerce (which in turn directly alters your overall commerce). Rivers for example produce a point of commerce for any tile adjacent to them being worked (unless it's covered by forest). Gold and Silver add lots of commerce. Coastal tiles add commerce. Finally, generally the biggest factor in increasing your commerce is building cottages. These will produce commerce for you, and as they are used, the will grow, eventually producing even more commerce. It's not unhead of to be getting 8 commerce from a single cottage (that has grown in to a town) in late game. Imagine having 6 or so of them in use around a town
Now, what does all of this mean? It means you want to specialise your cities! Given that the builings increase your commerce by a percentage, to maximise their effect, you want to maximise the commerce a city produces. Of course, a city maximising its commerce is obviously not going to have a high production, so to make up for this, you will want other cities specialising in production. In Civ 4, cities that do a bit of everything aren't the way to go.
