What is 2KHz mode in betaflight?
Does it give your quadcopter better performance?

I am new at this as well, but my understanding is that 2khz mode samples your Gyro sensor 2000 times per second. This allows the reduction of your PID loop time, in theory, to 500us. The closer we get to real time sampling the less important good PID values are because you no longer need to predict where the quad will be next.

With 2kHz you don't need any filters cause the data jumps (time jumps) are shorter. Becausea filters add delay, so betaflight is trying to rid of this. Plus some advanced scheduling get's your delay down to an minimum.

Is 2khz possible with F1 based FCs?
And how to check if 2khz is off or on?

Boris says yes F1 boards can run 2Khz mode, but I think it would be unreliable.

You need the latest betaflight to get the 2khz option, You must turn off ACC and Baro, and set gyro_lpf = OFF
Be aware you will only have 2 AUX channels left.

For F1 boards, 2khz mode is achieved by over-clocking a flight controller, basically it means the i2c bus is over-clocked, to get the gyro samples faster...

Check "Cycle time" in the configurator's status bar. If you get 500 that means 2kHz mode is on.

2KHz is from code writers thinking a bit too hard and constantly thinking that faster is better when there is much more to it than just that. There is negatives as well as positives to going faster and all the while you are getting diminishing returns with bigger headaches. I am good result at 1Khz mode, and tried 2Khz mode that didn't give me any improvement to be honest.

The gyro data doesn't seem to be good enough any more when sampling this fast... we need to get away from the MPU60XX or 90XX to something better, but that will just makes it more expensive.