The correct answers are You can choose either a cat or an owl., Both the cat and the owl are excellent choices., and I think I want both of them!

"Either" pairs with "or" to present a choice between two things.

"Both" pairs with "and" to include two things together. "Both of them" is also a correct pronoun phrase.

"Neither" must pair with "nor", not "or" (Neither the dragon nor the griffin...).

You cannot use "neither" with a negative verb like "cannot," as this creates a double negative. You should say, "You cannot choose either of them."