Yes.
real numbers
In a certain sense, the set of complex numbers is "larger" than the set of real numbers, since the set of real numbers is a proper subset of it.
the set of real numbers
The derived set of a set of rational numbers is the set of all limit points of the original set. In other words, it includes all real numbers that can be approached arbitrarily closely by elements of the set. Since the rational numbers are dense in the real numbers, the derived set of a set of rational numbers is the set of all real numbers.
Real numbers are a proper subset of Complex numbers.
Are disjoint and complementary subsets of the set of real numbers.
The set of real numbers is a subset of the set of complex numbers. For the set of complex numbers, given in the form (a + bi), where a and b can be any real number, the number is only a real number, if b = 0.
There is only one set of Real numbers.
No. A real number is only one number whereas the set of rational numbers has infinitely many numbers. However, the set of real numbers does contain the set of rational numbers.
Both rational numbers and integers are subsets of the set of real numbers.
Your question is ill-posed. I have not come across a comparison dense-denser-densest. The term "dense" is a topological property of a set: A set A is dense in a set B, if for all y in B, there is an open set O of B, such that O and A have nonempty intersection. The rational numbers are indeed dense in the set of real numbers with the standard topology. An open set containing a real number contains always a rational number. Another way of saying it is that every real number can be approximated to any precision by rational numbers. There are denser sets, if you are willing to consider more elements. Suppose you construct a set consisting of the rational numbers plus all algebraic numbers. The set of algebraic numbers is also countable, but adding them, makes it obviously easier to approximate real numbers. Can you perhaps construct a set less dense than the set of rational numbers? Suppose we take the set of rational numbers without the element 0. Is this set still dense in the real numbers? Yes, because 0 can be approximated by 1/n, n>1. In fact, you can remove finite number of rational numbers from the set of rational numbers and the resulting set will still be dense in the set of the real numbers.
It is the set of Real numbers.