Is "semisimple" a dense condition among Lie algebras?

The answer to the question in the title is "no". Semisimplicity is an open condition; however, it is not a dense open condition. Indeed, the variety of Lie algebras is reducible. There is one equation which nonsemisimple and only nonsemisimple Lie algebra structures satisfy, namely, that the Killing form Tr(ad(x)ad(y)) is degenerate, just as stated in the question. But there is also a system of equations which all semisimple Lie algebra structures satisfy, as do also all reductive and nilpotent Lie algebra structures, but solvable Lie algebra structures in general don't. These are the unimodularity equations Tr(ad(x))=0 for all x in the Lie algebra. These mean that the top exterior power of the adjoint representation is a trivial representation of the Lie algebra, which is obvious for any Lie algebra that coincides with its commutator subalgebra. But the nonabelian 2-dimensional Lie algebra is not unimodular. Hence in any dimension n, the direct sum of the nonabelian 2-dimensional Lie algebra with the abelian (n-2)-dimensional Lie algebra does not belong to the Zariski closure of semisimple Lie algebras.

I studied deformations of Lie algebras a lot and was able to construct a moduli space and a universal family for subgroups of an algebraic group:

http://www.mpim-bonn.mpg.de/preblob/4183

(I am very sorry to seemingly promote my work, please be sure I am not aware of any similar reference.)

I think you are interested by Lie algebras at large, while I can only give you informations on algebraic subalgebras of the Lie algebra of an algebraic group— i.e. subalgebras that are Lie subalgebras of a subgroup, the terminology is of Chevalley. You might however find some these results interessant or useful.

In front of all, this simple but striking consequence of a deformation theorem of Richardson—and the classification of semi-simple algebras:

3.18 COROLLARY Let $\mathfrak{g}$ be a Lie algebra and $k \le \dim\mathfrak{g}$. The map from the variety of $k$-dimensional subalgebras to the set of isomorphism class of semi-simple Lie algebras sending a Lie algebra to the isomorphism class of its semi-simple factors is lower semi-continuous, in the Zariski topology.

(Isomorphism classes of semi-simple algebras are ordred by injections.)

I cannot totally answer your question (for deformations of subalgebras):

“If the answer is no in general, is it possible to (nicely) characterize the Lie algebra structures that are in the closure of the semisimple part?”

However it is worth noting, that all what you need to get the description you are looking for, is to understand degenerations on nilpotent algebras. Indeed, consider a group $G$ and two subgroups $H$ and $M$ such that $M$ is in the closure of the conjugacy class of $H$. (The closure is taken in the moduli space, but replacing connected groups by their Lie algebras, you can embed the closure of the orbit of $H$ live in an appropriate Grassmannain variety.) So you have a curve $C$ in $G$ such that $M$ is in the closure of the translated of $H$ under elements of $C$. A consequence of the corollary is that if $M$ contains a semi-simple group $S$ then you can replace $H$ by a conjugate containing $S$ and assume that $C$ is in the centralizer of $S$.

So there is a natural way to enumerate possible degenerations, by looking at possible degenerations of the semi simple part (in the Levi-Malcev decomposition) and then study what can happen when you fix the smallest semi-simple part. In practice, small rank semi-simple examples should be very tractable, because we know very well the centralizers in the large group of maximal tori of the small semi-simple part.

A few words about the moduli space. The moduli space is not an algebraic variety but rather a countable union of (projective) algebraic varieties. This unpleasant phenomenon is indebted to tori: let me explain what happens for tori and for one dimensional subgroups of $SL_3$.

Tori. Let $T$ be a torus of dimension $r$, its Lie algebra ${\mathfrak t}$ contains a lattice $M$, the kernel of the exponential. As we know, the only $k$-dimensional subspaces of ${\mathfrak t}$ that are algebraic algebras are spanned by points of $M$. Hence the set $\mathfrak{P}_k(T)$ of $k$ dimensional subgroups of $T$ is the set of rational points if the Grasmann variety of $k$. Its topology is not exactly the topology induced by the Grassmann variety: it is on each irreducible component—which in this special case just means that $\mathfrak{P}_k(T)$ is discrete.

$SL_3$. Let me describe the set of $\mathfrak{P}_1(SL_3)$ of all one-dimensional subgroups of $SL_3$. We are then looking for $1$-dimensional algebraic subalgebras of the $\mathfrak{sl}_3$. There is the two nilpotent orbits, and the countably many orbits of $SL_3$ through $\mathfrak{P}_1(T)$, once we choosed a torus $T$ (these orbits almost never meet, by the normalizer theorem). Each of these orbits contains a nilpotent orbit in its closure, so from the smallest nilpotent orbit, you can generalise to any semi-simple orbit: subgroups in two distinct semi-simple orbits are abstractly isomorphic but not isomorphic as subgroups of $SL_3$.

Closing remark. I do not know enough scheme theory to write my work in a suitable language, so you may be puzzled by some terminology I use. I am using algebraic varieties as in Shafarevic I or in the book of Hanspeter Kraft (Geometrische Methoden der Invarianten Theorie). It is probably very clumsy! I am quite confident that these results would survive in the world of schemes, there is actually a scheme-theoretic version of Richardson's theorem, that Brian Conrad once showed me.

I would need more time than I have at present to try to write down a reasonable answer, but one quick comment on the question of deformation of representations, is that there is this paper by Nijenhuis and Richardson where they study the question of deformations of homomorphisms, hence in particular of representations, of Lie algebras and Lie groups. Nijenhuis and Richardson have several seminal papers on the deformations of Lie algebras, which are probably worth a look.

The inverse process to deformations is that of Lie algebra contractions, as in the original paper of Inonu and Wigner. These are limits which allow you to go from semisimple Lie algebras to their nonsemisimple "boundary".