Movement is bad. Panning in any form is probably a no-no. If you are working on a presentation that will also go to tape, you might shoot a well-composed pan and a series of still shots. Use the pan in your taped presentation. Use the stills in your streaming presentation.
Zoom lenses create a feeling of power for anyone that holds a camera.
But just because you can do something doesn't mean you should. Use the zoom to compose your shot. Then start the tape. Then keep your fingers off the zoom. Find your shot, then lock it down and record.