Can a mid wing airplane have no wing spar running through the fuselage?
Yes it can, albeit at the cost of some added weight.

By using reinforced bulkheads, the bending loads can be transferred into the bulkheads which leaves the center section free of structure. That is how large engines and their intake ducts can be placed where normally the spar would go. A classical example is the Lockheed F-104 Starfighter.


Lockheed F-104 Starfighter cut-away drawing (picture source).

Using bracing like on the Mini-MAX reduces stresses in the carry-through, but does not remove the need for something to keep the wings apart. In other words, the compression member of the wing spar is still required to transverse the fuselage where the wings are attached. Also, at high angle of attack the wing produces substantial nose thrust which could fold the wings forward were it not for a second compression strut at the wing nose. 


Mini-MAX 1100 (picture source)

In case of the Cri-Cri, which is a low wing design, the pilot sits on top of the spar. Moving the wing up to a mid position would create a much heavier aircraft and would require a wider fuselage, too.


 Colomban Cri-Cri (picture source)