If you try to lay a topcoat over an inconsistent surface, the inconsistencies will "print through" and give an inconsistent result.
Look at what you have there, you have a dog's breakfast of colors and surfaces. Different colors, stripper-etched vs normal, stains, and whatever that caulking or filling material is. (you're in trouble if it's ...
Primer is generally used for one or more of these reasons. If they apply, use it. If you're unsure, use it.
Save cost (cheap sealing and color/texture uniformity coat)
Improve adhesion over glossy or dirty surfaces
Well, they're new enough that they won't have lead.
The latex and other water based paints may have a problem, though. Nothing can grow in the oil based products, but very bad fungus and mold can grow in latex paints. It can smell like stinky paint in the can, but when you paint it, the stench won't dry out and will never go away.
This has been very ...
Yes, you can re-roll that area. Just use plenty of paint and blend it outward in a radial pattern so you aren't left with hard edges. Matte finish paint is quite forgiving, which is why we use it on ceilings.
I wouldn't sand it. Sanding latex paint can make things worse by tearing up the surface too much. ;
If it hasn't spoiled like Harper mentioned, there are some places that will recycle good latex paint. Otherwise, the can can be just opened and dried out. Then tossed in the trash. You can buy powders to speed up the process.
Oil based paints are considered hazardous waste. Check around your area, Best to find a Haz Mat disposal day ...
Another pic showing the overall door trim profile might help but the close up is valuable. From the little section of the wall that I see, if it were my house I would razor knife the white paint at the point where the trim meets the wall. If it falls away.... good. If it doesn't, maybe the next step may clear it off. Which is to remove the raised line of ...
Is it reasonable to just cut it away with a utility knife and then spackle over?
Yes, except for instead of spackle most painters would use a paintable water-cleanup caulking. You simply lay a thin bead of caulk along the crack, smooth it with a damp sponge or rag, then paint over it.
If you wanted to prime the ceiling for the purpose of blocking moisture that is a good idea. You can still prime it as long as the paint he used is not peeling from the surface. Or you could buy better paint made for bathrooms and places that may get moisture from steamy showers.