Current probes and Hall effect question
1) Current probes and non-contact current sensors are not necessarily just Hall-effect devices.  Some are hybrid Hall-effect/current transformer devices or even flux-gate/CT hybrids.  So the Hall sensor works for DC and low frequencies, then the current transformer takes over for higher frequencies.  Sometimes these are closed-loop schemes to eliminate gain error in the Hall-sensor.

2) Hall sensors are subject to many errors.  Residual field requiring degaussing, offset and gain errors, positional errors, stray field errors.  If all of these things are controlled or calibrated they can be very accurate.  My opinion is that a shunt resistor is typically capable of being more accurate, though it can dissipate power, drop voltage and of course is not non-contact.

3) See 1.