Be sure to scroll down... there may be more than one question on this page!
Spray acoustic ceilings are often applied over raw wallboard with no primer whatsoever. If this is the case in your home... it should be pretty obvious since the "gray" color of the wallboard should show through any residue from the spray texture... priming is essential. Though some of the paint-type texture products (I assume that is what you are going to apply) say they don't need a primer, I think this is an unnecessary risk on your part.
If the ceiling has been primed... again it should be obvious... and as long as the painter damp-wipes the ceiling down to remove all dust there is no reason to prime again.
If the ceiling is going to be textured with wallboard compound for a "deeper" texture, priming is an option but not as important since wallboard compound is much more "adhesive" that paint-type textures. After all, it is designed to stick tenaciously to dry, dust-free wallboard, right?
Judging by the difference in quality of today's construction compared to forty or fifty years ago and the much lower skill level required by today's workers, so-called "old school" workers still have a lot to teach today's overly cost-conscious contractors AND consumers. Unfortunately, many contractors are forced to compromise the total quality of their work to satisfy a consumer that is too concerned with the bottom line and less with the final product. This downward spiral of low price/ low quality / dissatisfaction has taken root in the home repair industry will be around until people begin demanding better quality WITH an educated willingness to pay for it. I for one prefer to be referred to as "old school" than as "progressive" if progressive means lower work standards.
If the ceiling is in otherwise good shape, just fill the texture with wallboard compound. If the ceiling is not a "flat" paint (meaning you can detect a glossy appearance to it), you might want to prime with a good primer sealer first before applying the compound. Bonding agents are for helping new cement to stick to old cement and are ineffective on painted surfaces.
Sand off any high spots in the existing texture, and then spread the wallboard compound over the entire ceiling using wide (8" or wider) wallboard knife. It will probably take a few coats to get the ceiling smooth. You do not have to sand between coats, but you should knock off any high spots or lumps with the wallboard knife (dry) before applying a second or subsequent coat. Sand smooth after the final coat. If there are still imperfections you can spot-fill them, again sanding after the compound dries.
This will be a huge job… no bout a doubt it… but me thinks less big than installing new wallboard. But… being the glutton for punishment that you are… if you think you would prefer the wallboard approach, use a 1/4" or 3/8" thick wallboard instead of the 1/2" size... you don't need the strength, so why bother! Plus it will be easier to install because of the lesser weight. Just use a construction adhesive to back up your screws (or nails).
You would have to take a sample of the ceiling material to a testing lab to have it analyzed for asbestos. Even if it is not asbestos, the dust you will generate will be enough of a pain to want to consider all other alternatives you suggested. If you are interested, read the article on asbestos at the website. Asbestos removal is not necessarily the first option. There are other ways to deal an asbestos problem... if you have it!
If you have enough ceiling height (and you seem to from your description) you can install a suspended ceiling. I am sure you have seen them... metal tracks that hold ceiling tiles. There are a wide variety of looks available, in both 2'x4' and 2'x2' sizes. They are fairly easy to install... probably one of the simplest and most satisfying home repair projects I can think of... and will solve all your problems at once... well, regarding the ceiling, anyway! My personal preference for high style are the 2'x2' tiles that protrude slightly through the tracks, giving an attractive depth to the ceiling.
Armstrong Products has good instructional material in their product packaging, as well as at their website, http://www.armstrong.com
If the lower portions of your ceiling are too low, start the suspended ceiling a slight distance up the slope. They you will only have to scrape off a more manageable amount of the texture. You can find or fabricate a decorative molding to cover the transition from the ceiling to the wall molding. This may be the most difficult part of the job! But the finished product will be well worth the effort. Besides the aesthetics, a fringe benefit is that you will save in AC costs... I noticed your address... by isolating roof heat to the area above the ceiling!
You shouldn't have to scrape down the entire ceiling… just repair the damaged area. There is an article on repairing textured at https://www.naturalhandyman.com/iip/infpai/popcorn.html. But before you can repair the texture you have to repair the loose wallboard tape.
Repairing the loose tape can be done in one of two ways. The first technique is to apply a small amount of wallboard joint compound on the ceiling where the tape is to be reset. Then press the tape firmly in place with a drywall knife, squeezing out most but not all of the compound. You will have to let this dry overnight and apply a second thin coat over the top of the tape. Once dry, sand smooth with 220 grit sandpaper, prime with an oil primer and perform the texture repair.
This method usually works but can cause a noticeable rise in the level of the seam or an uneven appearance that may even show through the texture. This is because it is difficult to press the excess wallboard compound out from around the tape, since it sits in the depression from the original taping. Rather than sand out the depression, try this trick... cut off the loose tape and install a new piece in its place. However, before installing the new tape, trim about 1/2 inch off the long edge of the tape. This gives you a little "wiggle room" in placing it in the wallboard compound. Apply a thin layer of compound and press the trimmed tape firmly into the center of the seam. Then apply a thin coat of compound over the tape. DO NOT OVERLAP THE OLD TAPE AT THE ENDS... it is unnecessary and may produce a noticeable lump! Let this mess dry overnight.
Knock off any lumps or high spots first with either sandpaper or a drywall knife. Apply a second coat of compound to level up the repair, extending or "feathering" the repair to at least 6 inches on either side of the seam. Sand the repair to a smooth finish after drying. Finish with a coat of primer followed by touching up the texture.
If the texture is not too deep, level and smooth the walls using drywall compound. It would be very difficult to REMOVE the texture... and some wall damage could occur... so using drywall compound is the better and less difficult way to go.
Use the first coat to fill in all the major depressions. A 12" taping knife is essential for leveling broad surfaces. The second coat refines the first coat. Use the taping knife... dry... to knock off any high spots from the first coat. Then fill in the defects left in the first coat.
After the second coat dries, you might want to do a little sanding with a 120 grit paper or sanding screen. A vacuum with a sanding screen attachment is the lowest dust option. Now apply the third coat of compound. This third coat may not even be necessary, or at worst should be a final leveling of any remaining unevenness in the wall.
Do a final sanding with a 220 grit paper or screen to get the walls flat. Vacuum thoroughly and wipe the walls with a slightly damp sponge... rinse frequently... to remove any residual dust and smooth in any scratches from the sanding.
Prime the walls with any high quality primer intended for drywall. If any stains or discoloration have appeared in the compound, use a stain killing primer such as Kilz. Then apply sizing over the primer before wallpapering.
There are special paint removers (try a paint store) designed to remove heavy wall textures, also, though I can't testify to their effectiveness from first-hand experience. If the texture is water based, you can also use a wallpaper steamer to soften it for scraping, though this method is admittedly iffy!