The end of WW2 hostilities was still some months away, so Bugs reminds the civilians in the audience not to do any unnecessary traveling. Gas, rubber, and spare parts for cars were rationed, and trains were needed to transport service personnel.
The pin ups Elmer sees are reminiscent of those drawn by noted artist Alberto Vargas, who produced many such pieces during WW2 for the benefit of U.S. troops.
"Is You Is or Is You Ain't My Baby" is a song by Louis Jordan. It was recorded in October of 1943.
"Oh, you and your old gun, you craaazy!" is an impersonation of Joe Besser, future member of the Three Stooges.
Bugs performs all three of what would become his signature moves here. He breaks the fourth wall several times, kisses his adversary, and appears in drag (though only partially here, when he wears lipstick).