4
Your approach is correct. In particular, sandwiching a controlled rotation between two CNOT gates is a common technique for implementing rotations on the $|01\rangle, |10\rangle$ subspace on hardware that does not implement it natively.
We can justify your approach using the fact that if $A$ has eigendecomposition
$$
A = \sum_i \lambda_i|i\rangle\langle i|
$$...
1
Yes.
A uniformly (Haar random) sampled state vector $|\psi\rangle$ is characterized by the fact that the probability measure is invariant under any $U$, i.e., colloquially, $U|\psi\rangle$ is just as likely as $|\psi\rangle$ for any unitary $U$.
On the other hand, a Haar random unitary $V$ is defined the same way: "$UV$ is just as likely as $V$, for any ...
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