When you trade bananas, do you want the skin only too?
Out of joke, I think one could trade an entire animal as well. For example, let's assume that if the trade is attached with an agreement of non cultivation/breeding, with a much lower cost, the animal is traded dead and in parts. Because that's not true that only the skin of a pig or sheap can be traded. Now if there is not this agreement, obviously the animal is alive and the cost is much more high. Of course the analogy is not true with Corn, Wheat and Rice, but it would still work in the same way with them. (= higher price without the agreement)
However, some ressources could be problematic like furs or dyes, sugar, whales, deers, etc... because they are very precisely the product of an animal or vegetable, or because they are savage animals that can't be breeded. Maybe the second could be "introduced" though, and the first just considered to be the animal/vegetable and not its product.
As to cultivating ressources in your empty squares, that's precisely the center of the idea: no need to trade with your neighbours in order to do so, you can start with your own ressources. Of course ressources growth capacity would have to be revised lower, for example 3 instead of 4 or 5, and not any other improvement could be build on it. One would have to choose between cottages or vegetable or animal farms.
The difference between vegetable and animal farms would be that vegetable ones would be harvestable right after their construction, but the animal ones would have to grow with time like cottages to reach their full efficiency. Both would have to be built near rivers until the discovering of the Well or the Bureaucracy. (increasing the strategic importance of rivers, too low in Civ4 IMO) Some animal would grow faster, some other would give more food, etc... each animal and vegetable would be different, with some possible exceptions, like one is like one other, but can't be harvested under the same climate.
EDIT: another idea about if a traded ressource can be grown or not, and that would make the agreement of no cultivation/breeding useless, is to have the same ressource into different states. For example, Wheat that can be grown, and Flour that can't be grown, Beet that can be grown, and Sugar that can't, Fur Animal that can be introduced, and Fur that obviously can't, Rice and...? Dried Rice? Corn and ? Pop Corn?

Maybe some ressources would have only one form as well. Maybe also that the different forms of a same ressource would have different effects.