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In general, the answer is 4, but only 2 of them are real. For example, the 4th roots of 16 are 2, -2, 2i, and -2i.
The 2 real roots are ±2.236 approx.If the domain is extended to complex numbers, there are also ±2.236i where i is the imaginary square root of -1.
The square of a real number is always a real number.
Yes a real number is a natural since real is natural so anything real is natural and so a real number is natural.
There is no difference between real solutions and real roots.