What do you use to attach gum paste figures to the side of a fondant covered cake so they don't slide?

What kind of "gumpaste figures" are you talking about? Are you talking about cutouts like stars, flowers, circles, etc. or do you mean something larger and heavier?

Use a tiny dab of buttercream or melted chocolate.

Debbie Brown's books alway say to lean the figure against the cake. It drives me nuts. I know it's not going to stay there. The first bit of movement and it will fall. I wish I had suggestions for you. I wait until the cake is delivered and set up before placing a figure on it.

Depends on the figure...cutouts can be attached by brushing water on them and attaching. Larger figures need RI, and I usually use toothpicks - I did this on a few cakes in my pics - Noah's Ark cake with an elephant head, ladybugs on a ladybug cake - just stuck one or two toothpicks into the figure depending on size (you may need to use more depending on the size of the figure) and then used that and RI to attach. HTH