Lobster - they are at odds if you are trying to have all you cities balanced - doing everything. Read the experts' articles around here about specializing a city for Great Person Production. Every other city should take advantage of Elizabeth's financial trait where applicable.
Irrelavant - anyone correct me if I am wrong please as I do not have the game in front of me, but anywhere you can build a farm, you can also build a cottage. Are you talking about before you get Pottery?
I try to hook up food resources and get adequate growth in a city before I build a cottage in it, but the cottages still need to be put up as early as possible on tiles where they will be worked. I see a lot about building them on river grasslands on these forums. And the experts here tend to be right while I am wrong, however, I have found that I get better results putting them equally on grasslands and plains, and farming grasslands and plains as necessary to keep growth up. To be more clear, if I have the food needed, I may have 4 grasslands cottaged, 4 plains cottaged, a couple grassland farms, and a couple plains farms, etc. assuming all the other tiles are special resources, hills, trees, coasts, or unworkable. Your mayors tend to like using the plains while building structures (if you have a food resource at 3+ food), and if they are on rivers with cottages, you pulling in 3+ commerce while working them to grow.