They say the Pegasus knights get +1 to their armor save because they are mounted. But does this rule apply to the characters riding Royal Pegasus or only to the PK?

Only the regular Pegasus Knights get plus one to their save for being mounted; a Royal Pegasus is a monster, and hence provides no such bonus.

Are pegasus knights skirmishers?

This question needs an answer, I think. I'd like to know it too. Do they have a 360° line of sight? Is there a -1 malus to hit them with shooting?

Pegeasus Knights are a unit of flyers and, like all flying units, skirmish. So yes, they do have a 360° line of sight and there is a -1 to hit them with shooting.