How much current draw from an inverter?
If I have a 12v car battery and I connect it to a 3kW 10A 220v AC
invertor, how much would be the current draw? 250 amps?

It may be more than 250 amps and it could be a lot less depending on how much load power you are taking.
If the inverter is rated at 3 kW this will be the maximum output power it can deliver. Given that an inverter might only be 90% efficient, the input power could be as high as 3.333 kW and then the current from a 12 volt battery would be 278 amps.
Of course, the inverter may have a surge power rating of 4 kW and then the surge current taken from the 12 volt battery might be as high as 370 amps.
But, largely speaking, the input current is determined by the load connected to the output of the inverter.