First off all im a beginner with electricity. I'd like to know at all times when the door is open (5 meters away) at all times, and have it displayed with an LED. What kind of sensor I'd need and how much would it cost? Also is there any way of making it as hidden as possible, so that there wouldnt be a bunch of wiring around the door.
I did something like this when I was a kid by connecting a wire to the part of the bolt mechanism that is inside the door and another to the metallic hole that it fits into when the door is closed. Connecting the LED in series with this circuit and a current-limiting resistor (if your doorknob and wiring don't already provide plenty of impedance) is a quick-and-dirty way to do what you're up to. Of course, this setup lights up the LED when the door is closed. You'll have to get a little more complicated if you want to reverse the logic.
Most interior doors are hollow and the wiring can easily be run inside of it. To hide the wiring for the doorjamb, run it under the moulding. A Dremmel with a cutting disk makes this an easy job. Paint over the wire once you've pushed it into the groove.
I wonder if my wiring is still in place in that old house. =]