Archie left his minstrel show to get to the hospital for Gloria's baby. An hour later, Gloria arrives. Then Gloria is in a hospital bed saying they've been waiting six hours. Barney Hefner arrives and meets Archie and Edith in the waiting room, and Barney is still in his blackface makeup from the minstrel show. There's no reason for him to still be wearing blackface seven hours after performing at the show.
In an earlier episode, "Mike's Pains," Mike was afraid to be in the delivery room, but later told Gloria he wanted her to see their baby's birth through his eyes, not a doctor's. However, when the baby is born in "Birth of the Baby: Part 2," Mike is standing behind Gloria as she delivers her baby, and she cannot see his eyes. (Most likely, Mike was following Dr. Shapiro's instructions).
Archie complains about the lousy parking at the hospital, and how it was impossible to find a parking space--but he doesn't own a car; on occasion he borrows Munson's cab.