This is a shot in the dark here, but are they all in
The Pride Lands from The Lion King?
I'm sure my responses are wrong here, but here it goes:
A deadly river (but not today)
Is this referring to the watering hole where Simba sings, "Oh I just can't to be King" and Zazu is almost killed? Or perhaps the Elephant Graveyard?
A monarchy ran by animals
The kingdom is run by Lions
A giant kitchen appliance
Is this referencing Pride Rock, which may resemble a giant can opener?
All three statements refer to a connected world. These are all in the same workd
Also, referring to one of the comments you made,
one of the answers accidentally got the right idea for the second line actually, which refers to Kant1's answer that a monarchy ran by animals was referring to the lion king.