False. For example, \(K_{3,3}\) is not planar.

True. The graph is bipartite so it is possible to divide the vertices into two groups with no edges between vertices in the same group. Thus we can color all the vertices of one group red and the other group blue.

False. \(K_{3,3}\) has 6 vertices with degree 3, so contains no Euler path.

False. \(K_{3,3}\) again.

False. The sum of the degrees of all vertices is even for all graphs so this property does not imply that the graph is bipartite.