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A two-fold question:
First of all, why is my bevel modifier not doing anything? Does not matter what values I put in.
And if I would add a bevel with Ctrl + B, the back edge of the shape is doing something different than the front edge as you can see in the picture below.
I just want to use a Bevel Modifier so I can change it later if necessary.
Secondly, would it also be possible to use a different bevel amount on the edge of the circle than the edge of the outside shape?
the file: 
First thing you have overlapping vertices, that's why the bevel doesn't work good, so select all and merge. Also, by default the bevel will be clamped if the bevel faces overlap with other faces, which will be the case here because the circular cavity is rather thin, if you push the circle on the Y axis you'll see that you'll be able to increase the bevel's width. If you want to keep the shape as it is you have 2 solutions, either bevel manually, or use the Limit Method > Weight of the modifier and determine the Bevel Weight value on the edges you want to bevel, it will allow you to vary the width:
