Like user2492 says, it's almost certainly moisture. Wood will warp and change shape as moisture levels change, and depending on the grain, each door may warp differently. Unless it's raising the bottom of the door for different flooring, or removing a lot of layers of paint, I leave shaving the door down as a last resort. Prefab doors are all designed with specific clearances and any mistakes are almost always from the installation.

To fix the hinges, you have two options, push things out or pull things in. Since you have to lift the door, you can push it out from the bottom and pull it in at the top. The most frequent issue I see is the door trim pulling away from the building frame at the top from the weight of the door. And the easiest way to pull that in is to remove the highest screw that is also closest to the hinge pin, and replace it with a 3" deck screw (assuming you have wood framing). The deck screw doesn't have threads in the last inch, so it will pull the hinge and door trim towards the framing when you get it all the way in. The reason for replacing the screw closest to the hinge pin is to also get the hinge to twist a bit and close tighter (compare the closed top and bottom hinge when the door is closed to see how the bottom hinge fits tight). Just be careful that you don't over tighten the deck screw and cause cracking or a lot of paint damage on the door trim.

Should you not be able to use a deck screw (e.g. non-wood framing), or for anyone that needs to push the hinge out, you can use a washer behind the hinge and return the screw through that washer. If you put the washer closer to the hinge pin, it will cause the hinge to stay open further when closed. And if you put the washer away from the hinge pin, it will cause the hinge to close tighter.

Yes, they tend to just be called Door Closers. They have a spring and a damper (oil or pneumatic) to slow the closing. They look like this:

Taken from http://www.doorstuff.co.uk/door-closers.php