is this ceiling drywall or plaster?
if it's drywall, then the texture is probably joint compound covered
with paint. the paint basically seals in the texture and makes it more
difficult to get rid of. in any case, getting rid of it is no small
matter. if you didn't have the crown molding, you could just hang new
drywall over it, but this wouldn't work with the molding i think. it's
possible there is some kind of stripper that would remove this stuff,
but it would be a ton of work and terribly messy. i think you might be
better off tearing the drywall off and starting over. if you scrape it
off somehow, you will spend many hours working over your head, and then
you will have a job of fixing the existing surface. those hours could
be spent on ripping out the old and redoing it.
if your ceiling is plaster, then things are more difficult yet.
i wouldn't hazard a guess whether there is asbestos there or not, since
i have no idea of the age of this ceiling and whether it is drywall or
plaster, etc. asbestos tests are cheap (usually less than $20) and
readily available. i have seen test kits in our local big box store.