As rhe title says: a normal number can contains itself?
From definition:
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc
For example if we take Champernowne constant (or $\pi$ or any other normal number) it's correct assume that at some point the number will contains itself?
No.
if this point presents, P = P* 10^N + Z (N and Z are integer) => P is the rational number.
Rational number cannot be a normal.