My issue is the following one :
When I try to bevel the selected edge, it "unbevel" another edge below
Any idea ?
https://file.io/agHRtkJ6QB8I
That looked weird for a moment in the screenshots, it was like what dark magic is this? But It's just that your model is very very small and the amount of bevel is clamped by the clamp overlap in the modifier, under the Geometry tab. Disable it clamp overlap and you'll get the normal behavior back.