Do molecules with bridges through rings exist?
I'm not sure about the existence of molecules with bridges through rings. However, there are several publications of synthesis of molecules mimicking wheels and axles ([2]rotaxanes; The “[2]” refers to the number of interlocked components) as one shown below (Ref. 1): 


(The diagram is from Reference 1)

This specific molecule (8; an “impossible” [2]rotaxane) represents a macro-cycle with a straight-chain molecule with bulky end groups going through its center. The inclusion of two bulky end groups prevents the straight-chain molecule leaving the macro-cycle (mechanically interlocked) as depicted in the diagram (See Ref. 2 for the total synthesis of the molecule).

Note that Ref. 1 also cited articles for the synthesis of [2]catenanes, which contain two interlocked rings (instead of one axle and one macrocycle). Keep in mind that there are some advanced catenanes and rotaxanes that exist (e.g., [3]catenanes and [3]rotaxanes).


(The structures are from Reference 1)

References:


Edward A. Neal, Stephen M. Goldup, "Chemical consequences of mechanical bonding in catenanes and rotaxanes: isomerism, modification, catalysis and molecular machines for synthesis,"  Chem. Commun. 2014, 50(40), 5128-5142 (https://doi.org/10.1039/C3CC47842D).
Jeffrey S. Hannam, Stephen M. Lacy, David A. Leigh, Carlos G. Saiz, Alexandra M. Z. Slawin, Sheila G. Stitchell, "Controlled Submolecular Translational Motion in Synthesis: A Mechanically Interlocking Auxiliary,"  Angew. Chem., Intl. Fd. 2004, 43(25), 3260-3264 (https://doi.org/10.1002/anie.200353606).