Is first cut hay better than second cut for horse?

Yes as the quality is better and horses are fussy about what they eat.

2) First of all there is no "correct" answer; the qualilty of any hay depends on many factors: weather, drying conditions, and at what point in the life cycle the grasses are cut, to name a few. An old farmer once described the difference to me as follows: first cut is like the main course; second cut is like dessert. They like the second cut better, much better, but the first cut has more body and fills their stomachs up quicker.