Yes.
When there is a current induced, there will always be some voltage however small this voltage might be.
We can have high voltage and small current. We can have low voltage and large current.
Both are however always present. If it is possible to measure one, then it should be possible to also measure the other.
No. For the simple reason that
you do not induce a 'current' into a conductor! You induce a
voltage. Any resulting current is determined by the values of the induced voltage and the resistance of any load connected to that conductor. If the conductor is open circuited, you still have an induced voltage, but no current will flow.