My old plug-in wall sconce just gave out on me. I have notived that most of the nicer looking sconces are all hardwired. Can I purchase a hardwired one and still have it work off of the light switch, and just change the outlet back to normal (rather than being turned on and off buy the switch)?

The short answer is yes. The actual wiring is slightly more complex than that. A switch simply interrupts the hot wire to open and close a circuit. That being said, you’d be well off to pick up a book from your home center before you proceed. It’ll have diagrams that will explain in detail exactly what you have to do.
Yes just remove the outlet and follow the directions on the new light.
Yes you can but if you are not comfortable doing it yourself get a electrician to do it.