Dichotomy Paper

I have a couple questions about the Dichotomy paper of Miller:

1) In Theorem 9 (the Perfect Set Theorem) there is a step that says to “recursively construct a continuous \psi:2^{\omega}\to 2^{\omega} such that (\varphi\circ\psi)[N_s]\cap(\varphi\circ\psi)[N_t]=\emptyset whenever t,s\in 2^n for some n. He says that you can do it because varphi|U is non-constant for nonempty open sets.

I’ve tried doing something like the schemes where we assign a clopen set to each node of 2^{<\omega} and assign to an x\in 2^{\omega} the intersection of the clopen sets for its inital segments. But I haven't really been able to make it work.

2) In the proof of the silver Dichotomy, we get a perfect clique and the homomorphism ends these to perfectly many equivalence classes. But I am unsure of why there must be a perfect clique.

3) Finally, for Lusin-Novikov, I'm not entirely sure why G_0 has to map into one vertical section. I know that each connected component has to go into one section, since we are applying a homomorphism. But the graph isn't connected (right?).

Help with these would be appreciated. Overall the paper looks interesting. I haven't been able to get into the rest of it as much I would have liked. But I think working through these examples will help when it comes to understanding the rest of the paper.