Yes
Only their own. In the case of an input note vp, that the nullifier is derived correctly from the contents of the note. In the case of an output note vp, that the commitment is computed correctly from the note.
Not exactly. configure
is called statically (see that there is no self
to have access to the witnesses or private values. synthesize
is called both statically to generate the verifying key (in this case we need to pass dummy values - see this explanation) and dynamically with the prover’s witnesses and public inputs