Before achieving escape velocity, does the object go around Earth in an elliptical orbit?
Not necessarily, but it may depend on how long an object is accelerating before it gets to escape velocity. If an object accelerates to escape velocity right away, it will just go off into an unbound orbit. A spacecraft might orbit elliptically for a while at velocities smaller than escape velocity, and later continue to an unbound orbit as rockets provide additional orbital energy. The exact trajectory will depend on many things.
