Here is a belated and perhaps too naive answer for your first question, i.e. a heuristic reason for why to believe in the Mumford conjecture.

One can think of cohomology classes on $M_g$ as characteristic classes for families of curves. That is, a class is the same thing as a rule such that whenever you are given a smooth family of curves $X \to B$ of genus $g$, this rule associates a cohomology class in $H^\bullet(B)$ in a functorial manner. (Topologically, you can also think of it as a characteristic class of oriented surface bundles of genus $g$, since $\mathrm{MCG}(\Sigma_g) \simeq \mathrm{Diff}^+(\Sigma_g)$.)

In a similar way, one might then think of a cohomology class on $M_\infty$ as a rule that assigns a cohomology class to a family of curves of *arbitrary* genus, or as a characteristic class of *arbitrary* oriented surface bundles. Of course this should be made precise since "functoriality" of such a characteristic class seems meaningless without any way of comparing surface bundles of different genus, and so one needs to define comparison maps between different moduli spaces by working with boundary components, gluing on tori/pants, etc. Nevertheless the intuitive picture is clear: a class on $M_\infty$ should be a rule that assigns in a uniform manner, to any family of curves $X \to B$ whatsoever, a cohomology class in $H^\bullet(B)$.

Now the $\kappa$ classes seem to fit the bill for being classes on $M_\infty$: given any $\pi \colon X \to B$ whatsoever, we may form the vertical tangent bundle, take its Euler class, multiply, push forward. This seems as canonical as one could hope for. Are there any others? Well, there are the $\lambda$-classes, i.e. the Chern classes of $\pi_\ast \Omega_{X/B}^1$, but Mumford showed via Grothendieck-Riemann-Roch that these are polynomials in the $\kappa$'s. It's hard to think of anything else.

Now surjectivity of $\mathbf Q[\kappa_1,\kappa_2,\ldots] \to H^\bullet(M_\infty)$ asserts that these obvious classes *are* really the only ones that you can write down in a uniform way, and injectivity says that there are no uniform relations between the $\kappa$'s (i.e. all relations are "low-genus accidents"). Now one might believe in the Mumford conjecture simply because people thought hard about these things and could not find any other genus-invariant characteristic classes, nor any genus-invariant relations between the $\kappa$'s. Mumford's conjecture is the simplest possible explanation for this failure.