Trying to wrap my arms around copulas

You probably haven't got an answer because everything you said is correct! As you said, Sklar's Theorem implies that everything about the "jointness" of the distribution is encoded by the copula, so once you know the $F_X(x)$ and $F_Y(y)$ and have a copula, then you know the joint cumulative distribution function.

$$H(x,y) = P(X le x quad & quad Y le y) = C(F_X(x), F_Y(y))$$

Or, if things are differentiable and you prefer to express things in terms of densities, you can write the joint density as

$$h(x, y) = color{red}{frac{partial^2 C}{partial x partial y}(F_X(x), F_Y(y))}f_X(x)f_Y(y)$$

where you can thing of the red term as the "correction" for non-independence.

(I originally posted this as a comment. While I think this is more of a comment than an answer, this interesting question has gone two years without any answers, and a short answer is better than no answer.)

Think of the density on the unit square as describing relationships between quantiles of the marginal distributions. If you look at a Gaussian copula for a high, positive correlation, when $X$ has a low quantile (say $0.1$), it’s likely that $Y$ will, too, and it is unlikely that $Y$ will have a high quantile. Ditto for when $X$ has a high quantile (say $0.9$).