Proving that the smooth, compactly supported functions are dense in $L^2$.

My intuition for (1) is to approximate the smooth, compactly-supported functions with compactly-supported step functions.

This is backwards. You should approximate things with smooth compactly supported functions, not the other way around.

I think there is no way around using mollifiers; how else will you construct smooth functions? Here is my approach:

Let $\Omega$ be any open subset of $\mathbb R^n$. For $x\in\Omega$, let $d(x)$ be the distance from $x$ to $\partial\Omega$. Given $f\in L^2(\Omega)$, define $$f_n(x) = \begin{cases} f(x)\quad &\text{ if }\ d(x)>1/n \\ 0\quad &\text{ if }\ d(x)\le 1/n \end{cases}$$ Note that $(f_n-f)^2$ converges to $0$ pointwise, and is dominated by $f^2$. By the dominated convergence theorem, $\|f_n-f\|_{L^2}\to 0$.

Next, approximate $f_n$ by $f_n*\phi_\epsilon$, where the mollifier $\phi_\epsilon$ is supported in a ball of radius $\epsilon<1/n$. This will ensure that the convolution is both smooth and compactly supported in $\Omega$. The $L^2$ convergence of mollified functions is a standard fact.

In advance: I don't have enough reputation to comment. This will not be a full answer.

I've worked through the density problem in a German professor's lecture notes.

You might actually get enough information out of them without really understanding German, since she relies heavily on symbolic notation. It's also proven for general $L^p$-spaces (except for $L^\infty$, of course).

These are the lecture notes: http://www.minet.uni-jena.de/~haroske/ha-1/ha-1_2.pdf‎

Look at page 46 ff.. There may be references to her first part of these notes, which you can get at http://www.minet.uni-jena.de/~haroske/ha-1/ha-1_2.pdf‎.

There are very few German words really, hardly a sentence, so any online dictionary might serve you well enough. The proof is quite elementary, you only need some knowledge about the Lebesgue integral as far as I remember.