Does dental anesthetic enter your bloodstream?

Yes, that's how: your body eliminates it, but should be slowly and small amounts

Yes: Anything injected into your body will get into your bloodstream...some more, some less...some quickly, some slowly. Local anesthetic will slowly be "washed away" by the small blood vessels in the area. A small amount of the anesthetic may also be broken-down locally at the site of injection. As the medicine is removed via the blood, your numbness will go away. Otherwise, you'd be numb forever.