"She does not run a lot."
あまり (amari) is used when you're saying you dont do something often, or not that much (for example not reading often, or not sleeping that much). たくさん (takusan) is used when you're saying there is a lot of something (for example saying there are a lot of books, or there are a lot of people.)
So basically you're saying that あまり is like the word "hardly" and たくさん is like the term "a lot"
If you leave kanojo out we're assuming that the speaker does not run alot. When in fact we're not talking about ourselves but someone else.
Duo wants you to talk specifically about a female third person, so you have to use it, but in context it wouldn't be mandatory