Both "I think that animals should not live in zoos" and "I don't think that animals should live in zoos" are grammatically correct. The placement of the negative is generally up to the author, though, and can be put in certain places within a sentence to create an emphasis.

Chinese grammar books sometimes, like to consider "preferrences" as "rules". There's so such rule to most natives. But you could say they sure like to say something some way "more often". The "think/believe" situation happens to rhetorical questions more often, though. And sounds natutrally-excusable.

Author's choice. A question of style.

动物不应该住在动物园里，难道要请到我们的餐桌上吗？ 呵呵

It's the choice of the author. Personally I believe the quotation from the book you mention is a stronger statement. Even stronger would be to omit the words "I think".