It is true that the double pendulum exhibits integrable behavior, when the initial angles are very small, however, in general, it is very difficult to characterize the chaotic behavior of the double pendulum in terms of the initial angles. There are other representations which provide a clearer picture of its chaotic behavior.
The introductory section of the following articleby O. A. Richter, and reference therein describe the main characteristics of the double pendulum integrability (see here for the official journal version). I'll summarize here the main facts:
(Remarks: The numerical values correspond to a standard double pendulum of unit masses and unit rod lengths and unit acceleration due to gravity)
The total energy of the double pendulum is a constant of motion. The double pendulum possesses 4 equilibrium points corresponding to total energies $E= 0, 2, 4, 6$. The total energy determines the topology of the energy hypersurfaces, for $E<2$, the energy hypersurfaces are three spheres, while for $E>6$ , the energy hypersurfaces are three tori.
To further understand this point, in the case of very low energies, the system can be approximated (linearized) to an isotropic harmonic oscillator. The energy hypersurfaces are then of the form $x_1^2+x_2^2+p_1^2+p_2^2=E = const.$, while for the case of very large energies, the kinetic energy dominates and we can neglect gravity. In this case, there are two types of solutions of the equation of motion, one in which the outer rod rotates and the inner rod oscillates, and the second in which both rods rotate. The transition between the two types of solutions is determined by the value of the total angular $L$ momentum which becomes a constant of motion (due to the lack of gravity). For the standard double pendulum the transition occurs at $ L^2 = 2E$.
Both limits (small and large energies) correspond to integrable systems. This is well known, but here is a short explanation.
To see that the isotropic harmonic oscillator is integrable, one needs to solve the equations of motion in polar coordinates.
The polar angle just rotates with a constant angular velocity, and the radial coordinates oscillates in such a way that the trajectory has the shape of a two trip course through the donut hole drawn on the surface of a two-torus. This is the Liouville-Arnold torus (whose existence indicates the system's integrability) with respect to which the three sphere energy hypersurface is foliated.
In the high energy limit, a similar Liouville-Arnold torus exists when the inner rod oscillates and a torus generated by the two polar angles when the two angles rotate.
(Here the exact solution is more difficult, see for example, the following article by Enolskii, Pronine, and Richter.
Now, since both limits of vanishing and very high energies correspond to integrable systems, the total energy also controls the system characteristics, but the dependence here is much more complicated.
The transition from integrability to chaos and back as the total energy decreases from infinity to zero is described in figure 2 of Richter's article. There are a lot of details, but here are the main features: The figure correspond to the projection of the trajectories
onto the plane spanned by the outer rod angle and the total angular momenta. For very large angular momentum, the projections of the trajectories are horizontal lines with constant angular momenta (which is a constant of motion). As the energy is reduced, two disjoint chaotic regions are formed, the integrable trajectories correspond to rational and irrational tori, together with stable resonances. At about E = 10.352 which corresponds to the golden winding ratio, all irrational tori vanish and a transition to global chaos occurs.
The stable resonances also vanish eventually, at the low energies the resonances corresponding to the second integrable region start to appear.
First. Yes, it is possible for the curves in Poincare section to cross. (I am assuming you mean generally). Remember, that Poincare section is a 2D projection of a 3D section of a 4D phase space. Regular, non-chaotic dynamics correspond to the winding of a $T_2$ torus embedded in this 4D space. Sections of such torus could be the couple of closed curves in the 3D hyperplane. Projecting this hyperplane on the 2D space could produce intersections. Additionally, Poincare section may include only points with a given directions of a hypersurface crossing, then the number of resulting curves would be reduced. Probably even then there still is a possibility for the intersecting curves in the Poincare section, although those would require highly nonlinear regimes (possibly after at least one bifurcation).
Second. Why would you expect the symmetry with respect to sign of the momentum? The Hamiltonian for the double pendulum contains the term proportional to the product of both momenta $l_\alpha l_\beta$. Therefore Hamiltonian and the torus in the 4D phase space are invariant only with respect to change of signs of both momenta,
but not with respect to a change of sign of a single momentum. After making cross-section and projecting it onto 2D plane, no mirror symmetry w.r.t. momentum sign is to be expected in the resulting Poincare section.
Addition Upon discussion in the comments, I understand that the one unresolved issue is the set of initial data which generates the figure from the book.
In the book the figure is accompanied by the comment:
To produce Figure 11.4 and all subsequent Poincare sections,
the initial conditions are varied. The total energy has the value $E = H(\alpha(0), 0, -\alpha(0), 0)$.
So, all those curves correspond to different sections arising from varied ICs with the fixed energy $E_0$. Here is my attempt at reproducing the figure:
Various colors correspond to an individual Poincare section defined by its own initial point. The point are chosen so that:
$$
\alpha=0, \quad \beta=0, \quad \dot{\alpha}>0, \quad H(0,l_\alpha,0,l_\beta)=E_0.
$$
The value of $l_\beta$ is chosen so that curves are evenly spaced between two limiting points corresponding to normal modes of oscillation. The last two conditions and known $l_\beta$ allow us to unambiguously calculate $l_\alpha$.
Finally, the limiting curve is defined by the equations:
$$
\alpha=0, \quad \dot{\alpha}=0, \quad H(0,l_\alpha,0,l_\beta)=E_0.
$$
The code for the picture is based on the linked question from Mathematica SE and can be seen here.
Best Answer
It's true that $k_{x,y}\to 0$ will be non chaotic, but, before this limit, the extra degrees of freedom of flexible arms should allow for more complex motion. At any rate I don't expect the degree of chaos to change too smoothly or monotonically with $k_{x,y}$, but rather that, e.g., periodic windows from resonances between spring and pendulum movements show up at intermediate values (resonances in the simple spring-pendulum have been studied for some time, see, e.g., here and here).
As Wrzlprmft comments, some simulations should help with building an intuition — it sounds like a nice undergrad project — if you do perform some, please share here your results. Edit: There are already a number of numerical implementations and simulation results available, for instance: in java (Open Source Physics), Matlab (1), Matlab (2), Mathematica (1), Mathematica (2), Maple, etc.