What was the “I’m not a bad slime” reference?
What was the “I’m not a bad slime” reference? A web search got me results regarding a game called Dragon Quest. Apparently, friendly slimes in that game greet players with 'I'm not a bad slime!' This is most likely a reference to that game.

In addition, I also found this, which says this is a reference to Dragon Quest. 

Regarding the air raids: From the link given by @AkiTanaka , a Japanese fan speculated that Shizue, who was summoned during the air raids in the 1940's, laughed not because she knew it was from a game or that she came from a time when the game already existed but because it reminded her of a friend, most likely summoned from a time where video games (especially Dragon Quest) already existed, who told her about this. Also, as @RossRidge said, the air raid was most likely the Bombing of Tokyo, which coincides with what is stated in Shizue's wiki page. 

Update: As @RossRidge mentioned, this was confirmed in Episode 8, with Shizue saying: 


   It was a school for kids from other worlds. ... It was one of them that taught me that video game line you quoted.