Is there a mistake in grammar?
Would "Tôi mở cuốn sách ngay lập tức" be more correct?
"Tôi mở cuốn sách ngay lập tức" is more natural, but both are equally correct.
The English translation is awkward and only relevant if the author meant to use in the present tense. For a better present tense sentence, "I am immediately opening the book."
Is cuốn= quyển
Yes it is the same