In terms of literature, I am not too sure where exactly one can see the proof, although I have seen $I(t)$ used in all articles related to the topic of infinitesimal 16th Hilbert problem. But maybe in the books written by some of the experts in the field?

Yes, it is exactly as you say: for small enough $\varepsilon >0$ the limit cycle is stable if $I'(t_0)>0$ and unstable if $I'(t_0)<0$. The function $I(t)$ conveniently encodes a lot of information about the qualitative behavior of the dynamical system in the relevant neighborhood: existence of periodic orbits as well as stability of the latter. Otherwise, to hope for semi-stability, assuming the dynamical system is at least real analytic, or even polynomial, one probably needs to have $I(t_0)=0, \, I'(t_0) = 0$ but $I''(t_0) \neq 0$. BTW, I've already fixed the $\ker$ typo. Thanks.

Well, there is some kind of an "analytic" formula for the solutions, but it is implicit and I don't know whether you need it: if my sloppy calculations are not wrong, $$\int_{A(0)}^{A(t)} \frac{d\tilde{A}}{D_0 \tilde{A} - a \tilde{A}^2 - a C_0 \tilde{A}^{\frac{a+b}{a}}} = t,$$ where the constants $D_0$ and $C_0$ can be expressed in terms of $A_0, B_0, X_0$. Analogously for $B$.

What I am trying to say is that one may get a metric in a very general context of polynomial vector fields, but this metric is rarely negatively curved.

I can only guess, as I haven't done any attempts to bring this idea any further, but one might be able to solve the system of two equations for the Christoffel symbols in more general situations, like higher degree Lienard equation for instance, since everything is in the ring of polynomials. However, do not forget that this system provides just a connection. After that one needs to show that the metric corresponding to this connection is in fact a metric (positive definite) and then check the inequality $K < 0$ which for general Lienard systems might not be satisfied.

I forgot to say that instead adding $-dH \wedge d\theta$ to your form, you can take any closed one-form $\omega$ on $E$ and take $-\omega \wedge d\theta$. Then again, the new form is closed and you have a symplectically perturbed map.

I see, so if the fibers are compact, then everything is ok. This means, as I expected, that the fibration is locally smoothly trivializible, by Ehresmann's theorem. By the way, you can obtain the fibration as a mapping torus of a more general smooth map, not necessarily a symplectomorphism. It requires only constructions from differential topology, without reference to symplectic forms. Of course, if you need your isotopy class to be symplectic, then you use the form. But in general, if I am not wrong, for a generic singularity, I think the gluing map is isotopic to a Dehn twist.

It looks like it. You can also have a one-parameter family, by taking $\varepsilon H$... By the way, since I do not know what kind of fibration you have, what its topology is and whether it is locally trivializable (something like Ehresmann's theorem), you have to make sure that the trajectories of the lifted vector fields make one full circle to return to the fiber $E_1$. If the fibers are compact, you have no troubles. Would you like to share some details about the fibration and what you need it for? There might be alternative ways for working with it...

Maybe it means that in some local coordinates the connection one-form is closed?! (could be component-wise?!) and pure gauge is basically that this one form is exact (it is the differential of a local section of the Endomorphism bundle of the tangent bundle?!)... I guess it's kind of like line-bundles in electrodynamics (the connection one-form there is the electromagnetic potential). Hence the "cohomology" reference... but I am guessing...