Well, I haven't seen the movie, but from the trailers and what I've read, the events that take place are not completely impossible. In the movie, what causes the natural disasters is a little more complex than just global warming. I'll do my best to explain.
Put simply, wind and weather are caused because the sun heats the earth unevenly. In an attempt to keep some sort of equilibrium, the earth has a couple of processes to counter the effect of the uneven heating. The first is wind, which works to transport warm air north and cold air south. The greater the difference in temperatures, the more work the wind has to do, and so winds will be stronger and we can expect more extreme the weather. The second process is ocean circulations, referred to more commonly as the Thermohaline Circulation. The Gulf Stream, for instance, is part of this circulation. Essentially, warm water is transported towards the poles, while colder waters are transported towards the equator (there are also changes in salt content that occur which is where the 'haline' portion of the name comes in).
Now Global Warming alone is expected to cause changes in temperatures and weather, which scientists usually talk of as occuring on the time scales of decades. However, there are some studies that suggest that Global Warming could actually lead to a complete shutdown of the Thermohaline Circulation, which is what the movie supposedly focuses on. If that were to happen, rapid climate change could occur more on the time scale of years versus decades. The poles would get colder and the equator would get warmer, and because the gradient in between would increase the winds would and weather would become more extreme (as I mentioned above).
Now the movie, goes one step further...suggesting that if the Thermohaline Circulation were to shut down, we could expect to begin to feel the effects in a matter of days. Now again, there are some studies (that I'm less familiar with) that suggest this could occur. There have been fossils of plants found in the glaciers in Greenland, suggesting that the ice formed on these plants so rapidly that they were preserved in ice before they even had a chance to die. This means that there was a nice warm period in Greenland, then suddenly a great snowstorm that buried these plants and afterward temperatures never warmed back enough to melt the ice.
I guess I'm a bit skeptical because I'm less familiar with the research that suggests major climate change could occur literally within days. And of course, not a lot of research has been done on that. So it's just a case of Hollywood jumping on an extreme. I mean remember back in the late 80's? There would people who would have you believe that by this time Killer Bees would be a national epidemic. Killer what? Yeah...whatever happened to that? I guess I just feel the same way about the way they are taking this climate scenario to the extreme. Sure it's possible...but is it really likely?
As for the earthquakes...I've got no idea. I did a little bit of searching on the web just now and didn't really find any articles from credible sources on the matter. But there seem to be a lot of mention that global warming would lead to an increase in volcanic activity and earthquakes. Without any background, I guess I would propose that if the poles were to get colder and the equator were to get warmer, it would cause more stress on the earth, causing tectonic plates to move around a bit more resulting in more earthquakes and volcanoes. I'm not sure if the explanation is really that simple though.
On a sidenote, there is some research that proposes that if more ice were to build up on the poles, the added weight would actually cause the tilt of the earth to change. I know there a small population of scientists that claim this is the cause of the last ice age. The poles of the earth shifted, for instance the northpole could've been the Northeast United States, and thus more ice built up there causing the ice age. Don't know why I thought of that...but just thought I would add that into the mix.
-Matt