In a random perfect matching of a regular bipartite graph, are all edges equally probable?
No.

Construction: Take two copies of $K_{3,3}$, one with the nodes $\{a,b,c\} \cup \{a&#39;,b&#39;,c&#39;\}$ and the other one with the nodes $\{d,e,f\} \cup \{d&#39;,e&#39;,f&#39;\}$. Remove the edges $(c,c&#39;)$ and $(d,d&#39;)$. Add the edges $(c,d&#39;)$ and $(d,c&#39;)$.