Does a Pokemon with the opposite gender from the trainer gain happiness or friendship faster?
The happiness of a Pokemon is not affected by the gender of a trainer. It can be affected in several ways, such as battling, fainting, or trading the Pokemon.
Note: Here is a popular belief of friendship:
You can read all about it here:
Just to clarify: Happiness and friendship is the same thing.
I'll probably add more information later; hope this helped!
Really short answer.