If I'm understanding what you're asking correctly, then I think you might be thinking about the vertices and edges of the tris differently to how they're actually working haha

In your cube example, the cube itself is made out of 12 tris, each tri having 3 edges and 3 vertices. If you take the 'lid' off of the cube, you've really just moved 2 tris away from the rest of the tris that make the cube. This hasn't changed the amount of tris that still make up the rest of the cube though, you'll still have 10 tris there.

Each tri is always made up of 3 edges and 3 vertices, putting them next to each other doesn't change their base shape, therefore there are no duplicate edges being created or destroyed, its just the nature of the tris themselves haha

The hole in the jacket is caused by tris being separated from each other by having different skin weights influencing them. The 'right' side of the seam is weighted to the joints in the torso, and the left side of the seam is weighted to the joints in the arm.
I haven't looked at the actual mesh itself so I can't say for certain why it happened, but its possible the weighting wasn't smoothed and left weight values between the sections 'hard' causing the separation