At one point Capt Marvel sees a large striped tiger, and goes off and investigates... even calling him 'mister'. In the original Capt marvel comics, Capt Marvel had a friendly tiger that talked called 'Mister Tawny'.
When Uncle Dudley reminds Capt.Marvel to switch Billy and he yells Shazam, there's a large collection of fire extinguishers against the wall. This stockpile is probably there to be safe from all the lightning bolts involved with changing personas.
The events depicted occur on September 10th, 23rd, and 24th.
Captain Marvel/Bully Batson are shown as living in Fawcett City. Captain Marvel was the most popular character of the Fawcett Comics publications, which were published in the 1940's.
Captain Marvel genuinely wants to hang out with the Team, but they misinterpreted it as needing to keep an eye on them. Pay in mind Billy doesn't use the JL's tech, so keeping an eye on them from the Cave wasn't an option.