Ceiling – Can you paint a ceiling that has acoustical texture?

Of course you can. Future owners of the house will swear at you to no end, but that's their problem, right? ;)

Priming then painting: The texture is water based, so you need to prime first with oil or shellac based primer. This will seal the texture and "waterproof" it.

Use plenty of ventilation and possibly an organic respirator.

Once primed, any paint can be used (make sure the full drying interval is used).

Removing: will work before any painting. Put down waterproof tarp, wall-to-wall and taped 6" up. Use a misting hand sprayer or a garden sprayer. Let soak 15 min, respraying if dry spots develop. Scrape with a long handled pole and a stiff angled putty knife. Keep the knife flat and start slowly until you get a rhythm. No sense in making another repair for gouges. Clean any residue with a new sponge mop and some TSP or a light detergent.