Turning the thickness display on is mostly visual, there is no physical overlap. When you turn thickness display off, you can see the middle stiffener just touches the plane of the L-bracket. The middle stiffener shares nodes with the L-bracket, so no contact is required.
The thickness display shows you mathematically that the element stiffness formulation will incorporate the offset. Bending is slightly affected by the offset. If the L-bracket was offset to the Top, then the center stiffener would not need to be extended. When the L-bracket is offset to the Bottom, the center stiffner is extended, which increases the mass of that bracket, so that is another mathematical effect of offset.
Yes, I started with surfaces that were on the same plane and customized the offset direction to get the visual appearance correct, which also provided a mathematically correct stiffness of the structure.
Since you are working in Explicit Dynamics where shell offsets are not supported, you have to use midsurfaces and contact to connect one shell to another across the two half thicknesses when the planes are parallel. You have to make sure the Pinball radius is large enough to cross that gap.