In a solid prone position the only movement you should notice at all is a slight vertical rise and fall of the sights from your breathing. Their should be now horizontal or angle movement at all. If you notice that type of movement something is in-correct about your position.
As far as sling tension, when match shooting my sling is very tight to the point where it is slightly uncomfortable but not painful. I was asked by a shooting coach once if my sling was "uncomfortably tight", I replied "Yes", to which he replied "Good your doing it right then". Before I talk much more about sling usage I need to ask how are you using your sling, are you using a "loop" style sling or using a "hasty" sling?
For the position of your elbows your non-shooting side elbow should be directly or as close as possible to being under the rifle. Some people cant do it but the more directly under the rifle you get it the better and the less horizontal movement you will notice in your sights. Focus on placing the non-shooting elbow and just let the shooting side elbow fall naturally to where its comfortable.
As for where in your shoulder you should place the rifle lift your shooting side arm in the air and feel your shoulder with the opposite hand. You should find that you created a pocket in your shoulder by raising your arm. You want to place the buttstock in this pocket. To try and fully seat the rifle in this pocket, when your getting in position angle the rifle up slightly with your shooting arm raised as as you place the rifle stock into that pocket, then lower the rifle and as you do rotate your firing arm down as well and it will kind of lock the rifle into place in your shoulder. If the whole buttpad/plate doesn't fit in the pocket then that is fine and not necessary. I would say only 2/3 to 3/4 of it will probably fit in the pocket with the rest of it sticking up above your shoulder line.
Also dont forget to bring your firing side knee up as far as comfortably possible. This will help stabilize your body and take pressure off your chest to minimize your breathing effect on your position.
Hope this helps!