On examples of what comics influenced the show:
The first season of the Adventures of Superman was based primarily on the radio broadcast shows from the 1940s. As Robert Maxwell produced both the radio shows and the first tv season, many stories were adapted from the radio broadcasts.
From Flights of Fantasy: The Unauthorized but True Story of Radio & TV's Adventures of Superman (a book which is by far the best resource I have found for information about this series).
The dialogue contained in Jor-L's speech to Krypton's Governing Council would be used again and again through the run of the radio show, whenever Maxwell felt the origin story needed retelling, and much of it would turn up in the first television episode, "Superman on Earth." - ch. 2
When it came time to develop Superman for TV Maxwell gave assignments to his writing staff from the radio days.
Ben Peter Freeman naturally wrote more episodes than anyone else; in some cases he merely had to dust off a few time-tested radio scripts. ch. 8
At the beginning of season two, with Robert Maxwell gone and Whitney Ellsworth as the new producer the focus changed.
Consequently, Superman's ploys veered away from the radio series towards the comic books. Stories that would appear on the newsstands around the same time they aired on television included "Panic in the Sky" ("The Menace from the Stars;", World's Finest #68) "Jimmy Olsen, Boy Editor" ("Jimmy Olsen, Editor;" Superman #86), "The Dog Who Knew Superman") ("The Dog Who Loved Superman;" Superman #88), "Lady in Black" ("Dick Grayson's Nightmare;" Batman #80) and "The Man in the Lead Mask" ("The Man Who Could Change Fingerprints;" Batman #82). "Five Minutes to Doom" focused on Superman's timely rescue of an innocent man from the electric chair, a theme that had appears in Action Comics #1. Only two stories -- "Jet Ace" and "The Face and the Voice" -- originated on radio. - ch. 12
Not withstanding Ellsworth's influence, many episodes were also original stories for the TV show. This was partly influenced by the writing staff. Jackson Gillis, whose background was in detective series, had little comic book influence, but he wrote many of the scripts for the show.
Freely admitting that his knowledge of the Man of Steel was "not more than reading the funny papers," Gillis likely viewed a couple of the 1951 shows as a primer, then went to work on his first assignment: "The Defeat of Superman." - ch. 12
There were comic influences throughout the remainder of the series.
Ellsworth made his annual sojourn to New York to discuss the year's budget with Jack Liebowitz and began conferring with Mort Weisinger on plots, a few of which would again come from comic book stories then in production. - ch. 13
TvTropes has this blurb:
The show was produced during the Mort Weisinger era of Superman comics, and he and Whitney Ellisworth were even on the staff. As a result, a lot of the plots are carbon copies of comic book stories from the time (for example, the story "The Phantom Superman" became the episode "Superman in Exile".) As a result, the series is probably the purest adaptation of late golden/early silver-age comic books out there.
Their actual example appears to be false, as the descriptions for "The Phantom Superman" and "Superman in Exile" do not at all line up.
To the second part
I wasn't able to find much of anything from the show that influenced the actual comics. The comics delved much more into science fiction and notable villains than the tv show. The TV show had very few scifi episodes. Most were Clark Kent mysteries of him trying to run down a story and solve a crime, only to have Superman come in the last scene to save the day.
