The first problem may be that you are cross chained. Your chain is traveling too far from inner ring to largest ring in the back. Maybe try shifting into your "granny" gear but shift up into a smaller cog in the back to help even out being in your small in front.
As for the chain jumping perhaps the derailleur hanger is bent slightly so when pedaling backwards the chain doesn't come off the casette straight and catches the rear derailleur. This should be an easy fix for a good bike shop that can straighten your hanger and tune up your shifting which would straighten out the chain and it wouldn't jump.