Those 260Hz Bandwith are due to its internal DLPF (Digital Low Pass Filter), and is not the sample rate.
Usually you use a BW filter that is half or less than your sample rate. But you can have, for example a BW filter of 20Hz and a sample rate of 200 Hz.
Going back to your questions.
Yes, the MPU6050 can have a sample rate of 1 KHz if you are able to read values out at that speed. It will be difficult if you are reading every gyro and accel every time and you are using I2C.
I think, not sure, there is a way to disable the Low Pass Filter, but I wouldn't do it, unles you are trying to read vibrations in the range of 260Hz to 500Hz (which is the maximum for a 1KHz sample rate).
Let me know if I can be of any more help.