At any particular speed, the current the motor draws will depend on the load on the motor. When unloaded the current will be low, and it will increase as the load is increased.
The inverter reduces the voltage applied at low frequencies, since the reactance of the windings is less. To a first approximation this is a linear function, eg at half the frequency the voltage would be halved, but many inverters are a bit more sophisticated and apply a little more voltage at low speeds to compensate for the fact that although the reactance of the windings is less, the resistance is the same. So if they did not compensate, the torque available would be less at lower speeds. This compensation should ideally be set up to suit the individual motor.
When you run at speeds over the normal maximum, eg 50 Hz, then the usual inverter cannot increase the voltage any more. So the available torque will fall off. This is not usually a problem, since the sort of things we do at maximum speed don't usually require high torque. So for instance I can run my little Unimat 3 at up to 8000 rpm, which is good for things like the little circular saw attachment. That means setting the frequency to 100Hz, and the motor runs at twice nominal speed.
The University of Canterbury tested ordinary production motors back in the 80's and found that they were OK for balance at up to 10,000 rpm, which is also about the speed that the bearings were rated for. So you could have safely gone to 150Hz with those motors, if you needed to go really fast. If your inverter could supply the right voltage, that would also give you three times the normal rated power. (Provided the insulation was also up to the increased voltage!) Apart from insulation limitations, there will also be an increase in eddy current losses as the frequency goes up, so you can't increase the frequency indefinitely. However, three times normal speed is certainly possible.
It is not just Ohms law when starting. The motor can be modeled as a transformer with a shorted secondary, but since the coupling between primary and secondary will not be perfect there will also be some leakage reactance. That also helps to limit the starting surge.
John