1. What you would be served in a "typical" Dominican household is known locally as "la bandera," ("the flag") because it is so common: rice & beans served with meat, often mixed together (in which case it's a "loquio"). But many other very Dominican dishes, such as san cocho (sorta, but not exactly, a stew), asopao (kindof a tomato-ey rice soup with seafood or meat mixed in), pica pollo, their version of spaghetti (everything mixed together, often with chunky vegetables included), pastelitos (little meat-filled pastries, sometimes with egg and/or raisins thrown in), and all sorts of things made with plantains -- mangu, mofongo, tostones (my personal favorite)....
As far as restaurants go, you can find many cuisines other than Dominican, esp. in the capital, Santo Domingo. Italian, French, Middle Eastern, Spanish, Chinese (although I only found one that I thought was good), Mexican (although I do not think the Dominican version of Mexican is very good), American-style steakhouse.... Can't find things like Thai or Indian, or for that matter, any of the other truly spicy-hot foods. Truly good seafood is hard -- but not impossible -- to find in a restaurant.
2. customs -- about what??? would help in answering this if you were a bit less all-encompassing in this question....
3. main sport, nay, the national passion, is BASEBALL. Basketball, volleyball, boxing are all popular, but nowhere near the level of popularity of baseball. And baseball's popularity is shared by the women -- not as much just a male thing as it is in some countries. There are other sports growing in popularity, such as bicycling. And of course the rich play golf, tennis, race boats and cars. But baseball is king.
Hope this helps.
Regards,
Keith