The description of orbits in the $d=4$ and $n=2$ is not difficult.
It is better to work projectively i.e. look at the orbits in the projective space $P(C^5)$ of nonzero binary quartics up to a constant. A nonzero binary quartic $F$ can be written as a product of linear forms and therefore corresponds to a collection of four points on the projective line with possible repetitions but no ordering. If all points are distinct one can transform them by an $SL_2$ element into 0, 1, $\infty$ and some other guy $\lambda$. Orbits in this case are in one-to-one correspondence with the $j$-invariant $$j(F)=\frac{4}{27}\times\frac{(\lambda^2-\lambda+1)^3}{\lambda^2(\lambda-1)^2}$$ Essentially $\lambda$ is the cross ratio of the four points but this depends on the chosen ordering. The above rational fraction is what is needed in order to kill the dependence on the ordering. The same $j$-invariant can be expressed in terms of invariants of $F$: $$ j(F)=\frac{S^3}{S^2-27 T^2} $$ where $S$ is the invariant of degree 2 and $T$ is the invariant of degree 3 (both defined up to normalization by a constant). They generate the ring of invariants you are talking about. What you said above is false: this ring is not generated by the discriminant which is not a fundamental invariant. It is given by the denominator $S^3-27 T^2$. Finally the other orbits correspond to three points or less. These points can be placed anywhere we want on the projective line by an $SL_2$ element. So these orbits are characterized by the multiplicities $(2,1,1)$, $(2,2)$, $(3,1)$ and $(4)$. The last two form the null cone of binary forms with a root of multiplicity $>$ half the degree of the form. All invariants vanish on these ones. What one needs to distinguish them are covariants: joint invariants of the form and an extra auxiliary point. For $n>2$ one needs mixed concomitants, i.e., joint invariants of the form and an extra auxiliary complete flag.