The cycle double covering conjecture states that every bridgeless graph admits a collection of cycles such that every edge in the graph is contained in exactly two cycles from the collection (allowing repetitions). The counterexample with the smallest number of edges is proved to be cubic. My first question is that whether it is true that such a minimal counterexample has the property that removing every edge yields a bridge. In other words, is it true that if $G$ is a graph with a cycle double covering, then $G\cup e$ also has a cycle double covering, where $e$ is an added edge between two vertices of $G$?

My second question is whether anything is known regarding cycle coverings that cover every edge the same number of times.