What is the right order for 16bit data on 8bit 8080 data bus
You really need to read the portion of the data sheet pertaining to 8-bit transfers yourself to understand this.

In RGB565 mode, it could be barely argued as "big endian" in the sense that the Green bits bridge a byte boundary and the more significant ones are sent first.

In RGB666 mode each color gets and is fully contained within its own byte transfer, and since there is no relative significance between colors the question does not apply.

With the exception of "memory write" and "memory read" most registers are only 8 bits, so it does not apply to those either.  For the memory commands, you may have to figure it out for yourself, if you use those.