For example in say FireRed. I trade an Abra for that Mr. Mime. Is there a chance that the Mr. Mime would be shiny?
No, Not from In Game Trading:( Serebii Says You Can only Get Shinies through wild pokemon, eggs, and the battle tower.
I'm pretty sure by reading this page http://www.serebii.net/games/shiny.shtml that shiny Pokemon can only be found/obtained in battles. Hope this helps. :)