I'm continuing back on forth on my old model and this one, trying to find where I went wrong. I used different rigging techniques or scripts, RapidRig, HumanIk and different positions of the joints, if that would be able to fix the problem, but no luck so far. At this point I created my own skeleton I want to try binding to the mesh and see if the same errors occur, as well as an ABAutoRig I want to test.
I tried different scripts for deformers, but I don't have enough knowledge about it and I couldn't find an easy answer (nSkinMorph/Shapes and scripts like those.)
Below I will post a bunch of screenshots of the different rigging methods with the same issue arising.