Is the elemental summons supposed to be different for different players? Mine shows red, but a couple in my alliance are showing yellow and purple.
Yes, that is the way it is supposed to be.
It is different for everyone and is dependant on what day and time you started playing E&P. Everyone starts the game at different times but the elemental color always starts with red.
Does it stay in the same order (except for special events)?
Yes, always the same order YPRBG (yellow purple red blue green)