The derivation of the quantum information no-hiding theorem

Question

I am reading Samuel L. Braunstein, Arun K. Pati, Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. I am puzzling over the derivation in Section Perfect hiding processes.

1. The authors have already assumed from the outset that only the ancilla $A$ and thus its orthornomal basis $A_k$'s depend on $\psi$. Does it not already imply their conclusion "any information about $|\psi\rangle$ that is encoded globally is in fact encoded entirely within the ancilla. No information about $|\psi\rangle$ is encoded in system-ancilla correlations (nor, in fact, in system-system correlations)."? There is a statement preceding it stating that "We may swap $|\psi\rangle$ with any other state in the ancilla using purely ancilla-local operations". But I do not see how and why this is necessary. Moreover, what would it be like to have "system-ancilla correlations"? Would it be for $|k\rangle$ to depend on $|\psi\rangle$?

2. Does Equation $(3)$ of the paper 

$$\alpha^*\beta\langle A_l(\psi)|A_k(\psi_\perp)\rangle+\beta^*\alpha\langle A_l(\psi_\perp)|A_k(\psi)\rangle=0 \tag3$$
come from the following requirements?
$$\big\langle A_l(\alpha|\psi\rangle+\beta|\psi_\perp\rangle)\big| A_k(\alpha|\psi\rangle+\beta|\psi_\perp\rangle)\big\rangle=0, \tag{3.1}$$
$$\big\langle A_l(|\psi\rangle)\big| A_k(|\psi\rangle)\big\rangle=\big\langle A_l(|\psi_\perp\rangle)\big| A_k(|\psi_\perp\rangle)\big\rangle=0. \tag{3.2}$$
If so, what is the rigorous rationale for Equations $(3.1), (3.2)$?