I don't think sanding the surface will produce a surface good enough to paint. It's certainly a lot of effort for uncertain results.
If the underlying surface is very rough it might still show through the wall paper, unless it's very thick (1000 gsm lining paper for example) or you do some sanding.
The best answer might be to continue to peel off the paper and then skim the wall with plaster. Depending on your plastering skills you might want to get a professional in.
The result should be good enough to paint.
If the ceiling feels spongy anywhere at all, some sort of deterioration has occurred. The most common cause is moisture seepage—whether from leaky plumbing, roofing, or pets—but plaster deterioration can also occur from sustained high temperature (120 °F/49 °C) as is common above an enclosed light fixture or electrical junction box carrying a heavy electrical load.
Decay could also be caused by certain pests like cockroaches, fleas, or larger critter's waste.
The first step to fixing it is to stop whatever is causing the problem. Then assess what needs to be done.
Wall and ceiling work is somewhat expensive if you hire someone to do it, but it is quite inexpensive if you do it yourself. Even if you don't know what you are doing and have to (re)do it four times, that will still be less expensive than hiring a pro. For a head start, visit your local Home Depot the next time they have a sheetrock demo. Or search for "sheetrock repair" on Youtube.
If investigation or repair require tearing down the ceiling, the price can escalate if you see something else you want to do: rewire, insulate, reroute plumbing, redo siding, etc. Frankly, the add-ons are the most difficult (financially and emotionally) aspects of such projects.
Best Answer
Google "textured painted ceiling" to find a wide variety of materials and tools including rollers with various textures. The edges were probably done last with a small roller that has no texture.