This is an excellent question. The short answer is that we don't know.

A major concern is that melting of the Greenland ice sheet will force the Gulf Stream - a current transporting tropical water from the Gulf of Mexico north to Iceland - south. This would have the consequence that northern Canada and Europe would suddenly get a lot colder, and may have been responsible for the Younger Dryas, a brief, ice age-like interval between 13,000 and 12,500 years ago. Ironically, human-induced global warming might actually make the Earth cooler!

However, the fact is that we don't know for certain. We can be pretty certain that events won't play out in the way that they do in the movie "The Day After Tomorrow".

1. Here's an answer to your first question:
If global warming results in increased rainfall in the North Atlantic, and the melting of glaciers and sea ice, the influx of warm freshwater onto the sea surface could block the formation of sea ice, disrupting the sinking of cold, salty water.

2. And here's an answer to your second question:
Ocean currents act much like a conveyor belt, transporting warm water and precipitation from the equator toward the poles and cold water from the poles back to the tropics. Thus, ocean currents regulate global climate, helping to counteract the uneven distribution of solar radiation reaching Earth's surface.