Asbestos in texture was infrequently used as a reinforcing fiber. A $25 test will verify whether it is a problem and is worth doing. Simply sample the texture using a film container or similar small clean container. An analytical lab will run a PLM test to identify and quantify asbestos content.
Once the ceilings are stripped there will be some rough spots and gouges. These should be filled with drywall topping compound and sanded flush. The ceiling can then either be finished smooth (painted), or textured. Hire a drywall contractor to apply texture. He has equipment and experience to do this and is well worth the money. There is no such thing as texture paint; well there is, but you don't want it.
The better your finish preparation (filling and sanding) is, the better the outcome will be. Texture is often used to hide an inferior taping job. The faults can be corrected, but it takes some time.