Question

of what category is the set of all integers in R and in itself.

Answer 1

A set is of first category if it can be written as a countable union of sets whose closure has empty interior. (A set whose closure has empty interior is commonly called nowhere dense, but I won't use this term in the remainder of this answer.) Otherwise it is of second category.

So, the set of all integers in R is of the first category and in itself is of the second category.