My parents had artex on every single wall downstairs in their house - god when I fell down the stairs once and my arm caught the wall on the way down, Christ I think I took half the skin off!!!
Better if you can get it off, but it can be a very messy business. Most plasterers can and will plaster over it, but before doing that it will need bonding/ sealing with PVA this is diluted with water, the reason for this to take place first is to stop all the water being sucked out of the plaster as its being applied, not allowing the plaster to dry slowly. Other than this the plaster will crack and fall off.
Yes. It's done all the time.
Some will do it teacake's way with PVA.
My own preference is to put a coat of "Carlite Bonding" straight onto the Artex. It acts as a backing coat. Then just apply "Carlite Finish" plaster, or more likely, "Gypsum Multi-Finish"
"Bonding" is what you use to plaster directly over ceramic tiles, or even glass.
Yes I've seen some very bad skimming jobs. It looks ok- ish during the day, switch a light on at night, oh dear humps and bumps. Looks like the tide coming in. :0)