The estimations are not based on numerics, but rather reasoning. For example, since real-valued polynomials are stable under complex conjugation, each complex root must have a complex conjugate. For polynomials with three distinct roots then, one must be real. This is just an EXAMPLE of the kind of reasoning that can be used to infer whether a root is purely real or complex.
As for formulas: no, there is no GENERAL formula for the quintic polynomial or higher. General formulas only exist for polynomials of degree four or lower.
Of course, there are SPECIFIC formulas for specific KINDS of higher-degree polyomials. But for the general quintic and higher, nothing. That's already been proved classically.
> As for formulas: no, there is no GENERAL formula for the quintic polynomial or higher.
Note that there is no general formula for quintics (and higher) in radicals.
But it is a bit arbitrary to draw the line there. There is also no 'general formula' using basic arithmetic operations (addition, subtraction, multiplication, division) that gives roots for x^2 = 2.
However it is quite useful to be able to talk about those roots, so we added a dedicated symbol for them (sqrt), and more generally extended the definition of exponentiation to fractional powers.
If it were generally useful to talk about the roots of quintics we also would've added common notation for their roots.
The roots of a polynomial vary smoothly as the coefficients vary. This is intuitively obvious. This is an important part of Abel's proof of the unsolvability of the quinitic. The probability of a random number being ambiguously real or complex has measure 0 (measure < epsilon, for all epsilon > 0), so you can ignore all computationally ambiguous cases. You don't need an exact calculation of any specific polynomial's root, to get the right answer of the measurement over all polynomials.
As for formulas: no, there is no GENERAL formula for the quintic polynomial or higher. General formulas only exist for polynomials of degree four or lower.
Of course, there are SPECIFIC formulas for specific KINDS of higher-degree polyomials. But for the general quintic and higher, nothing. That's already been proved classically.