Questions from a current regulating and measuring circuit
I'd change that to switch the high side. 


You do not want the wire sitting live when you don't need it to be.
It let's you measure the voltage across the wire single ended.


I'd use a separate micro-ohm shunt resistor to measure the current though, some thing that does not get so warm and change resistance with current. Or for this, a different sensing approach that is not so sensitive to the heat.

I'd also drop that frequency lower, sub 100Hz, even then that wire will sing a bit, but it won't be as noticeable as your current ~1kHz.

That op-amp is not rail to rail either, I'd suggest a LMC6482 or similar.

Zener should be fine provided your op-amp drive is less than the zener current.