Hello Arai san,
1. As an alternative to the AT93C66A, you can use 8-TSSOP version CAT93C66 whose datasheet can be found from the following link: https://www.mouser.com/datasheet/2/308/CAT93C66-D-69419.pdf
2.) As mentioned in the datasheet, GX3 Supports serial EEPROM (93C56/66) for storing USB Descriptors, Node-ID, etc.
A Serial EEPROM is different from I2C EEPROM. There is not a necessity to use beyond 4K Serial EEPROM for GX3.
3.) No, GX3 uses the serial EEPROM to store the USB descriptor and other device-specific info to enable the plug-and-play usage of the device.