J.S.
What we think may be not so good is completely normal!! Babies use their mouth to explore their surroundings and learn about things. She is just starting to realize what she has control over and is testing her limits (and yours too!).
Not much to do about throwing herself backwards, just try to keep her safe as possible. One year olds are often covered with bumps and bruises though.
Have you bought teethers that are a harder surface like a crib? Perhaps try teething tablets or tylenol before putting her down.
Throwing stuff out is a favorite pastime of babies. How many times will Mommy pick that up? Just leave everything on the floor for a little bit-don't pick it up immediately. This is the very beginning of her learning cause and effect. And of course, keep soft toys that areen't dangerous if thrown.
Just follow your gut and you'll be fine.