For Bott-periodicity reasons $0 \to C_0(\mathbb{R}^2) \to C(S^2) \to \mathbb{C} \to 0$ is also important. The generator of $K_0(C_0(\mathbb{R}^2))$ comes from the Bott projection associated to to Hopf line bundle on $S^2$. Tensoring by $C_0(\mathbb{R}^2)$ performs the "double suspension".

Here's a counterexample: en.wikipedia.org/wiki/James%27_space

Do you know of a good reference for your second statment? I'm curious as to what happens if you go from a totally disconnected attractor to a connected attractor by continuously varying the IFS.

For a general measurable function $k : [0,1]^2 \to [0,1]$ the integral you have defined may not converge. If, on the other hand, $k$ belongs to $L^2([0,1]^2)$ then you get a Hilbert-Schmidt operator.