The Bethe equations and Bethe ansatz are the central tools to study quantum integrable systems and  there are thousands papers devoted to them. I cannot pretend to know substantial part of this research, so let me write some remarks which I am aware of.

You ask  "... is there **logic** to their roots ... " 

Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory). 

Let me quote from [String hypothesis for gl(n|m) spin chains: a particle/hole democracy][1]
Section 3.1 page 10.

> Suppose that N is large and some Bethe root $\lambda_n$ has a positive
> imaginary part. Then the l.h.s of (26) is exponentially
> large with N. To achieve this large value on the r.h.s. there should
> be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in
> the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using
> the reality of solution of the Bethe Ansatz [36], we conclude that the
> Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

> where s is an integer. These complexes are called **s-strings**.
>
>  String
> hypothesis in its strong form states that all solutions of the Bethe
> Ansatz equations can be represented as a collection of strings, and
> that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N
> precision. In its strong form the string hypothesis is wrong. However
> there is an evidence that its weaker version is correct if the proper
> thermodynamic limit is taken. The weaker version states that most of
> the Bethe roots are organized into strings with exponential in N
> precision, and that the fraction of solutions which significantly differ
> from (27) decreases to 0 when N → ∞. We discuss in more details
> applicability of the string hypothesis in appendix A.

  [1]: http://arxiv.org/abs/1012.3454