They are called real numbers. Negative square roots must be complex numbers.
A number with a square root is still a number. So the answer is all of them.
The square roots of a number are numbers and, as numbers, they are not capable of doing anything.
Any negative number, and infinity. However, a number (called "i") was created to represent the square roots of negative numbers.
There is no specific name but they comprise the set of non-negative numbers other than perfect squares.
Every whole number
They can, but they are called imaginary numbers. This is because a "square" of something is the number that, when multiplied by itself, will equal the first number. A negative multiplied by a negative equals a positive, and positives can't turn into negatives on their own, so there are no square roots of negative numbers.
Rational numbers whose square roots are whole numbers are themselves whole numbers. They are called square numbers, e.g. 1, 4, 9, 16, 25 and so on.
Numbers with square roots that are whole numbers
They are called perfect squares.
A square root is not a number system. Square roots of non-negative numbers may be rational or irrational, but they all belong to the set of real numbers. The square roots of negative numbers do not. To include them, the number system needs to be extended to the complex numbers.
Some irrational numbers are square roots, but not every one. Not every square root is irrational.