June's hairstyle varies quite a bit during Season 6. In general, it was changed from a long-ish "flip" to a short cut, as seen in the opening credits. (Other changes for this season include a jazzy arrangement of the opening music, which had been the same for the first five seasons, all suggesting the producers might have been worried about lackluster ratings. This season, number 6, ended up being the last one.) As for June's hair style, the long-ish "flip" is seen in the first three episodes, and number 6 (Eddie, the Businessman (1962)) while episode 5 (Double Date (1962)) features the short cut. In this episode (Number 4 of Season 6, we may be seeing both: in the opening scene, it's short - though it may just be pinned back for housework - while a short time later, when Ward comes home from work, it's long-ish "flip" again.
When Beaver confronts what he thinks is the paper boy, the sidewalk and street appear wet. It's not that it had rained or even that this was an effort to simulate that it had rained; the sidewalk and street were hosed down with water as a production technique used to mitigate the glare that those surfaces would otherwise have emitted and, consequently, interfered with the lighting effect desired for the scene.
Cleaver home phone number is given in this episode as KL5-4763. The number of the local newspaper office, as reported to Ward by Beaver, is KL4-6287.