My brother is tiger dob 7/5/1962 , & his girlfriend is horse dob 13/9/1978. Are they compatible?

Oh, yes. They make a pefect match. They are doomed to be together for life long.

For auspicious dates, May 5, May 21, June 8, June 20, July 7, July 12 and July 21 are for their choices! Personally, I recommend May 21 most, because it is really full of luck in Chinese culture.