1) what is the conventional way of computing a face normal?
I know this is done using the crossproduct, but the resulting vector depends on the definition-order of the vertices (Clockwise or Counterclockwise). So i was wondering...
The D3DXComputeNormals routine computes face normals too, but there is no parameter to specify the correct definition-order you want to use. So this function assumes you to deliver your faces defined clockwise or counterclockwise. What is the commonly used definition order?
This is somtimes a tricky topic; there is not a standard for this; the two main graphic API directx and Opengl are different in that by default; and several CAD programs uses different coordinate system; in some you define faces in closewise order, some in counterclockwise; in some Y positive mean going up, in some Y negative mean going Up; heck, in some Z coordinate mean Up/down and Y coordinate is used for depht.

That is why several import models tools dialogs includes options for invert Y or Z values, or swap YZ planes when reading from different file format models.

For Directx I think it is in clockwise by default for defining front faces; Y positive mean going UP; and the camera should be aiming to the positive Z plane;

If you calc a normal passing the vertex order, v1,v2,v3 you will get the same result when you calc the normal passing the order v3,v2,v1 BUT THE RESULT IS INVERSE SIGNED. This mean the calc for computing the normal is exactly the same whatever what coordinate system your are using; just the result have to be interpreted different; the normal (0,0,-1) could mean pointing front in one coordinate system, or could mean pointing back in another coordinate system.

Dont ask which coordinate system is better or which you should use; i dont think there are adventages in one agaist the other.

2) Some things about the crossproduct are not yet clear to me. Okay.. it can be used to determine the plane normal. But does it point up or down??. Most websites say that you must use the right hand rule to determine the normal direction, but doesn't this depend on the handedness of your coordinate system
The normal dosent just point Up or down; it point perpendicular to the plane; in a flat floor face into a room it point UP; in a flat ceiling face it point down; in the flat vertical left wall in the room it point to the righ; in a slanted floor it could point diagonal. it dosent matter the handedness of your coordinate system your are using as long you pass the vertices order accordilly to that system when you calc the normals.

If you are geting procedures/functions downloaded from internet which involve passing normals to it then YES; you have to be carefull to pass the normal calculated using the handedness of the system expected by the rutine;