There are shallow water wells all over the Gulf of Mexico, including much of the American Gulf coast. Go to the beach in Texas and you can see them dotting the seascape.
I believe that Florida does not allow them along their shores, because the tourism industry there is stronger than the oil industry.
The Western half of the Gulf is pretty heavily tapped, but the eastern Gulf has some fairly promising oil reserves that haven't been explored. Some of these are in deep water, which was too expensive to drill in when oil was cheap. Oil prices need to stay high for these reserves to be commercially viable.
Many Americans believe (with help from politicians) that the government was holding the oil industry back from exploring these reserves, but the real obstacle was cheap oil from the Middle East. Why drill if you can buy it for less from the Arabs?
BTW, what they pump into a well to cap it is concrete, which has to be mixed differently for different depths, probably due to the different temperatures. A mile down, the water can be below freezing temperature. Apparently one thing that went wrong on the BP well is the wrong mix of concrete was pumped down to the well.
Mud is the nickname for the mixture of fluids pumped down through the drill and out the drill bit to lubricate and cool the bit, and pump drilling debris up and out of the well.
CD