That benign looking bamboo grass that has started in the corner of your yard is
the grass equivalent of kudzu, very invasive. Known as Japanese stilt grass,
it's botanical name is microstegium. It can be pulled rather easily now but
don't wait too long since it seeds in late August/September and seeds are
viable for seven years. This article will give you lots of info:
http://www.nps.gov/plants/alien/fact/mivi1.htm

As for plants for the slope, you may have problems since many plants don't like
to grown under/around roots of walnut trees.