You may have to use patching material and sand it well. Be careful sanding, that does not appear to be sheetrock in your ceiling. Primer the patch, before painting, and dont put tape around like that anymore. If you practice a bit, painting a ceiling line, isnt really very hard at all.

It’s like a weird paper material. It happened all the way around this room.

Should I use a spackling compound on the ceiling?

Also, can I use any primer or is there a special ceiling primer?

Yes you can use spacklng compound. I would use a good quality sealing bonding primer, and you may have to repaint the entire ceiling so patches do not show. No, ceilings do not take different primers, but, since you are patching a lot, a sealing bonding primer would work best. Make sure you let the spackling dry well, and try to match in the texture of the ceiling, or it will still show bad, after primer and painting. Its probably bigger job than you think. Your ceiling is some type of wallboard, it appears, or maybe ceiling tile that have had their grooves filled and repainted. Its not the look of sheetrock. Do not use tape this time, please. If you dont think you can do it right yourself, please call someone to do the spackle and texture work, and then you can just primer and paint yourself.

Use this primer first -- it's made to seal torn drywall.

Use joint compound, not spackle to repair.

You can use the Gardz to prime the repair, then repaint.

Plenty of how to videos on YouTube. Here is a link to a channel with some good ones.

https://www.youtube.com/@vancouvercarpenter