Is it possible to power a microcontroller via inductance from an AC power line?
No, the wire will not radiate a magnetic field along the lengt of the wire but straight out of the wire. To make a transformer and extract energy you need an magnetic field that alternates. As you will have a split core to measure the current the best idea would be to use it to both measure the current and extract power to your circuit. This only works with a real current transformer and not with a modern Hall effect variant, if you need the Hall effect model for accuracy in your measurements you will need a separate split core current transformer to power your circuit.

The resultant circuit is much like an transformer with only one turn of primary winding, the current in the secondary winding will be the current in the AC wire divided by the number of turns in the current transformers winding. Note that the circuit outputs a current and if the load is not sufficient the voltage can be quite high, therefor you need to be sure to load the current transformer and to have a over voltage protection that can handle the excess power, especially in events of high current on the AC line, like starting currents of an electrical motor.

If you then put a resistor or other current sensing element in series of the output of your current transformer that measures the total current you can know the current trough the AC line at the same time as you uses it to power your current sensing circuit.